The mathematical world

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The mathematical world

Photo by James Clarke

Some philosophers think maths exists in a mysterious other realm. They’re wrong. Look around: you can see it

James Franklin is professor of mathematics at the University of New South Wales in Sydney. His book An Aristotelian Realist Philosophy of Mathematics is out this month.

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What is mathematics about? We know what biology is about; it’s about living things. Or more exactly, the living aspects of living things – the motion of a cat thrown out of a window is a matter for physics, but its physiology is a topic for biology. Oceanography is about oceans; sociology is about human behaviour in the mass long-term; and so on. When all the sciences and their subject matters are laid out, is there any aspect of reality left over for mathematics to be about? That is the basic question in the philosophy of mathematics.

People care about the philosophy of mathematics in a way they do not care about, say, the philosophy of accountancy. Perhaps the reason is that the certainty and objectivity of mathematics, its once-and-for-all establishment of rock-solid truths, stands as a challenge to many common philosophical positions. It is not just extreme sceptical views such as postmodernism that have a problem with it. So do all empiricist and naturalist views that hope for a fully ‘scientific’ explanation of reality and our knowledge of it. The problem is not so much that mathematics is true, but that its truths are absolutely necessary, and that the human mind can establish those necessities and understand why they must be so. It is very difficult to explain how a physical brain could do that.

One famous philosopher who finds mathematical necessity an inconvenience is Peter Singer. In one of his best-selling books on ethics, he argues that we cannot rely on intuiting ethical truths, since the most convincing case of intuition, in mathematics, is not correct. ‘The self-evidence of the basic truths of mathematics,’ he says, ‘could be explained… by seeing mathematics as a system of tautologies… true by virtue of the meanings of the terms used.’ Singer is wrong to claim that this philosophy of mathematics, called logicism, is ‘widely, if not universally accepted’. It has not been accepted by any serious philosopher of mathematics for 100 years. But it is clear why anyone who, like Singer, wishes to explain away the strange power of human intuition might want a deflationary philosophy of mathematics to be true.

To the question: ‘Is mathematics about something?’ there are two answers: ‘Yes’ and ‘No’. Both are profoundly unsatisfying.

The ‘No’ answer, whose champions are known as nominalists, says that mathematics is just a language. On this view, it is just a way of talking about other things, or a collection of logical trivialities (as Singer claims), or a formal manipulation of symbols according to rules. However you cut it, it is not really about anything. Those whose encounter with mathematics at school was less than happy (‘Minus times minus equals plus/The reason for this we need not discuss’) might feel some sympathy with the nominalist picture. Then again, it is also a view that appeals to physicists and engineers who regard serious propositions about reality as their business. They look on tables of Laplace transforms and other such mathematical paraphernalia as, in the words of the German philosopher Carl Hempel, ‘theoretical juice extractors’: useful for getting extra sense out of meaty physical propositions, but not contentful in themselves.

Nominalism might have a certain down-to-earth appeal, but further reflection suggests that it can’t be right. Although manipulation of symbols is useful as a technique, we also have a strong sense that mathematics makes objective discoveries about a terrain that is in some sense ‘out there’. Take the subtleties of the distribution of primes. Some numbers are prime, some not. A dozen eggs can be arranged in cartons of 6 × 2 or 3 × 4, but eggs are not sold in lots of 11 or 13 because there is no neat way of organising 11 or 13 of them into an eggbox: 11 and 13, unlike 12, are prime, and primes cannot be formed by multiplying two smaller numbers. The idea is very easy to grasp. But this doesn’t mean there’s nothing to discover about it.

It turns out that the way in which the primes are distributed among numbers involves a complex interplay of pattern and irregularity. On the small scale, the latter is most evident: there are long stretches without any primes at all – indefinitely long stretches, in fact. At the same time, it is widely believed that there are infinitely many ‘prime pairs’; that is, pairs of numbers only two apart that are both prime, such as 41 and 43.

When we turn to the large scale, the impression of disorder fades and a pattern starts to emerge after all. Primes become gradually less dense as one counts up: the density of primes around a large number is inversely proportional to its order of magnitude. The density of primes around a trillion (1012), for example, is about half what it is around a million (106). More exact information on the intricacies of the distribution of primes is contained in the Riemann Hypothesis, currently the most famous unproved conjecture of mathematics.

It seems as if pure mathematics reveals the topography of a region whose truths pre-existed investigatio, even language

This is typical of the results of pure mathematics, from simple school facts such as the divisibility of numbers by 9 if the sum of their digits is divisible by 9, up to the higher reaches of abstract algebra. It is impossible to escape the conclusion that pure mathematics reveals to us the topography of a region whose truths pre-existed our investigations and even our language.

Inspired by that thought, Platonism proposes a philosophy of mathematics opposite to nominalism. It says that mathematics is about a realm of non-physical objects such as numbers and sets, abstracta that exist in a mysterious realm of forms beyond space and time. If that sounds far-fetched, note that pure mathematicians certainly speak and often think that way about their subject. Platonism also fits well with the apparent success of mathematical proof, which seems to demonstrate how things must be in all possible worlds, irrespective of what the laws of nature might be in any particular world. The proof that the square root of 2 is an irrational number does not rely on any observationally established laws. It shows how things must be, suggesting that the square root of 2 is an entity beyond our changeable world of space and time.

Still, despite its clean lines and long history, Platonism cannot be right either. Since the time of Plato himself, nominalists have been urging very convincing objections. Here’s one: if abstracta float somewhere outside our own universe of space and time, it’s hard to imagine how can we see them or have any other perceptual contact with them. So how do we know they’re there? Some contemporary Platonists claim that we infer them, much as we infer the existence of atoms to explain the results of chemistry experiments. But that seems not to be how we know about numbers. Five-year-olds learning to count don’t perform sophisticated inferences about abstractions; their contact with the numerical aspect of reality is somehow more perceptual and direct. Even animals can count, up to a point.

In any case, the problem with Platonism is not so much about knowledge as about its view of mathematical entities. Surely when we measure, or calculate, or model the weather mathematically, we are dealing with mathematical properties of real things in this world, such as their quantities. Such properties are not abstracta: like colours, they have causal powers that result in our seeing them. The visual system easily detects such properties as the ratio of your height to mine (if we stand next to each other). There is no room for abstracta in other worlds to enter the story, even if they did exist.

Nominalists and Platonists have fought each other to a standstill, each convincingly revealing the fatal flaws in their opponents’ views, each unable to establish their own position. Let’s start again.

Imagine the Earth before there were humans to think mathematics and write formulas. There were dinosaurs large and small, trees, volcanoes, flowing rivers and winds… Were there, in that world, any properties of a mathematical nature (to speak as non-committally as possible)? That is, were there, among the properties of the real things in that world (not some abstract world), some that we would have to recognise as mathematical?

There were many such properties. Symmetry, for one. Like most animals, the dinosaurs had approximate bilateral symmetry. The trees and volcanoes had an approximate circular symmetry with random elements – seen from above, they look much the same when rotated around their axis. The same goes for the eggs. But symmetry, whether exact or approximate, is a property that is not exactly physical. Non-physical things can have symmetry; arguments, for example, have symmetry if the last half repeats the first half in the opposite order. Symmetry is an uncontroversially mathematical property, and a major branch of pure mathematics – group theory – is devoted to classifying its kinds. When symmetry is realised in physical things, it is often very obvious to perception; if you have an asymmetrical face, don’t go into politics, because it makes an immediate bad impression on TV. Symmetry, like other mathematical properties, can have causal powers, unlike abstracta as conceived by Platonists.

Another mathematical property, which like symmetry is realisable in many sorts of physical things, is ratio. The height of a big dinosaur stands in a certain ratio to the height of a small dinosaur. The ratio of their volumes is different – in fact, the ratio of their volumes is much greater than the ratio of their heights, which is what makes big dinosaurs ungainly and small ones sprightly. A given ratio is something that can be the relation between two heights, or two volumes, or two time intervals; a ratio is just what those relations between different kinds of physical entities share, and is thus a more mathematical property than the physical lengths, volumes and so on. Ratio is what we measure when we determine how a length (or volume, or time, etc) relates to an arbitrarily chosen unit. It is one of the basic kinds of number. As Isaac Newton put it in his uniquely magisterial language: ‘By Number we understand not so much a Multitude of Unities, as the abstracted Ratio of any Quantity, to another Quantity of the same kind, which we take for Unity.’

Any digression into applied mathematics – rarely undertaken by philosophers of mathematics, who prefer the familiar ground of numbers and logic – will turn up, for the alert observer, many other quantitative and structural properties that are not themselves physical but can be realised in the physical world (and any other worlds there might be): flows, order relations, continuity and discreteness, alternation, linearity, feedback, network topology, and many others.

There is a name for a philosophy of mathematics that emphasises the way in which mathematical properties crop up in the actual world. It is called Aristotelian realism. It is based on Aristotle’s view, opposed to that of his teacher Plato, that the properties of things are real and in the things themselves, not in another world of abstracta. A version of it, holding that mathematics was the ‘science of quantity’, was actually the leading philosophy of mathematics up to the time of Newton, but the idea has been largely off the agenda since then.

Infants and animals demonstrably do have the ability to recognise pattern and estimate number, shape and symmetry

Because Aristotelian realism insists on the realisability of mathematical properties in the world, it can give a straightforward account of how basic mathematical facts are known: by perception, the same as other simple facts. Ratios of heights are visible (to a degree of approximation, of course). Infants and animals demonstrably do have the ability to recognise pattern and estimate number, shape and symmetry.

Our developed human intellectual abilities add two things to those simple perceptions. The first is visualisation, which allows us to understand necessary relations between mathematical facts. Try this easy mental exercise: imagine six crosses arranged in two rows of three crosses each, one row directly above the other. I can equally imagine the same six crosses as three columns of two each. Therefore 2 × 3 = 3 × 2. I not only notice that 2 × 3 is in fact equal to 3 × 2, I understand why 2 × 3 must equal 3 × 2. So the Platonists were right to call attention to the ability of the human mind to grasp mathematical necessities; they just failed to notice that those necessities are often realised in this world. The second intellectual ability by which the human mind extends the results of perception is proof. Mathematical proofs chain together a series of insights, individually similar to ‘2 × 3 = 3 × 2’, to demonstrate necessities that cannot be understood at a glance, such as how the density of primes tails off for large numbers.

Aristotelian realism stands in a difficult relationship with naturalism, the project of showing that all of the world and human knowledge can be explained in terms of physics, biology and neuroscience. If mathematical properties are realised in the physical world and capable of being perceived, then mathematics can seem no more inexplicable than colour perception, which surely can be explained in naturalist terms. On the other hand, Aristotelians agree with Platonists that the mathematical grasp of necessities is mysterious. What is necessary is true in all possible worlds, but how can perception see into other possible worlds? The scholastics, the Aristotelian Catholic philosophers of the Middle Ages, were so impressed with the mind’s grasp of necessary truths as to conclude that the intellect was immaterial and immortal. If today’s naturalists do not wish to agree with that, there is a challenge for them. ‘Don’t tell me, show me’: build an artificial intelligence system that imitates genuine mathematical insight. There seem to be no promising plans on the drawing board.

The standard alternatives in the philosophy of mathematics have failed to account for the simplest facts about how mathematics tells us about the world we live in – nominalism by reducing mathematics to trivialities, and Platonism by divorcing it from the world, the real world of which mathematical truths form a necessary skeleton. Aristotelian realism is a new beginning. It connects the philosophy of mathematics back to the applications that have always been the fertile ground from which mathematics grows. It has a message both for philosophy and for mathematics and its teaching: don’t get blinded by shuffling symbols, don’t disappear into a realm of abstractions, just keep an eye fixed on the mathematical structure of the real world.

Read more essays on logic, mathematics and metaphysics


  • Sean

    I suspect that many philosophers will basically agree with this article. I think a similar view of mathematics is offered in Bede Rundle's book, "Why There is Something Rather than Nothing".

    • James Franklin

      If you'd like to summarise or quote something brief from Rundle, I'm sure people would be interested.

      • Sean

        Hi James, thanks for the reply. Like you, Rundle seems to emphasize "the way in which mathematical properties crop up in the actual world". He sees numbers, for example, not as Platonic entities but as a kind of property of objects. Some quotations:

        "the ultimate explanations of statements of number are given with the applied forms, as when we say that '10 is even' may be explained as 'Ten things are equal to twice some number of things.'"

        "... 1,000,000 is greater than 1,000, not because the former is physically larger, but because a million of anything is more than a thousand."

        "You can add an apple to a pile, or take one away. That is addition and subtraction as operations on concrete things, and addition and subtraction of numbers is parasitic on this procedure, on adding or taking away numbers of apples, and so forth."

        (These quotations are taken from Chapter 6 of the book mentioned above).

        • James Franklin

          Interesting. It sounds close to the older Aristotelian theory that mathematics is the "science of quantity". What is the point of it in the context of Rundle's book?

        • HaakonKL

          x = -1

          1000 × x > 1 000 000 × x

          This is sort of trivial to disprove isn't it?

  • templeruins

    I think what Mathematics tell us, that many western philosophers overlook, is the inter-dependancy of all things. Things are only definable by others things. Things are always in a state of flux. Mathematics may provide a bridge between ultimate and conventional truths.

    Plato set western philosophy back, so far back we are still recovering. Plato's "inherent existence" set people off on delusions that mathematics actually has always been showing us otherwise. Plato's "ideal forms" has been an anathema to mathematics philosophical study to this day. It's a shame we haven't been looking what has been staring us the face for this long.

    Take what a revelation Einstein's e=mc2 was. This interdependency of apparently separate and demarcated forms was near sacrilegious to many scientists of the day, even to Einstein himself! This is a vestige of the "ideal forms" holding us back so long even though mathematics has been showing us the way in an opposite direction for hundreds of years.

    You concept of a "neat" "box" of eggs is entirely subjective and has no "truth" to anything in reality. 11 or 13 eggs could be arranged in plenty of ways to fit into a box, regardless of whether you find their organization "neat".

    Savants who can do extraordinary calculations are not tapping into some ethereal dimension of mathematics, like it is popularly portrayed. Their minds are exaggerations of the unexplained human brains confluence with pattern making. Anyway, that's another thing, I'll leave it at that.

    I'm sure there's some connection to Plato's world of forms and the thinking behind Descartes' dualism.

    • Hominid

      Einstein's formula is consequential NOT because it's mathematical, but because it's descriptive of a relationship.

      Math in and of itself is the ultimate delusion - a meaningless abstraction of something that doesn't exist. It has meaning only when the unit is defined. The difficult and meaningful task is describing the unit.

  • Ingolf Stern

    Mathematics has no subject, just like word-language has no subject.
    Math is the language of relationships.
    Language is not what we talk about, it is how we talk about things.

    • James Franklin

      The article gives reasons why that view of mathematics is false, although it's commonly held by outsiders.

      • conchis

        Perhaps it's because you were simplifying for a lay audience, and/or I'm not sufficiently clear on what nominalism actually claims, but I found the reasons given in the article rather unclear/unconvincing. "Mathematics must be about something because we have a strong sense that it makes objective discoveries about things" sounds less like an argument than an assertion - but I'm sure I must be missing something here.

        • James Franklin

          The assertion was followed by examples of what mathematics is about, objectively. What more do you want?

          • conchis

            I'm clearly missing the connection between the examples and the argument - which may be because I'm dim, rather than because the argument is unclear. I'm not trying to be difficult here; I'm genuinely confused.

            Perhaps it's because I lack a sufficiently clear understanding of what it means to say "X is objectively about Y", and what the implications of the truth or otherwise of that statement would be, but neither the fact that mathematics can say internally consistent things about the distribution of primes, nor the idea that thinking about primes can illuminate egg-carton design strike me as strong arguments that mathematics is "objectively about" anything.

          • James Franklin

            I don't really understand what "internally" is doing in what you write. If it's about primes and primeness is found in egg carton arrangements, what's "internal" about that? If biology is about e.g. facts about the circulation of the blood, what would be the point of adding that those facts are "internally consistent"?

          • G

            I'm with you, or perhaps I'm just persuadable in my ignorance looking for answers. Clearly the objects of mathematics exist in nature: number, symmetry, and so on. Once we start from those objects, the relationships among them follow with the necessity of mathematical proof, which is absolute. Whereas my beloved empiricism can only offer 'support' for hypotheses, within a specified range of error, but never 'proof.'

            Thus it necessarily follows that those mathematical relationships also exist in nature, though at a deeper or at least more subtle or less obvious level. This, just as 'meaning and purpose' exist in human minds, and since minds exist in brains, and brains exist in nature, so do meaning and purpose exist in nature. In a way, relationships are as close as anything can get to being 'nonphysical,' even though they are measurable. A ratio is measurable, but it exists as a relationship rather than as a property of this or that object. (Frankly I'm somewhat uncomfortable with the word 'nonphysical,' suggesting as it does something otherworldly rather than something immanent.)

            In the end this comes very close to mysticism, properly defined as the search for direct encounter with the ground of being. Much mumbo-jumbo has hitchhiked on that word, but in its pure form it seeks to understand the relationships between nature and those uncomfortably nonphysical and sometimes measurable aspects of reality. One could go on a recursion loop over all this, but perhaps that's just a Western-tradition equivalent of a Zen koan.

            In the end the goal is comprehensive understanding, and its ethical correlate is comprehensive empathy and compassion. Those latter properties don't appear to map to anything mathematical, but they do map to qualia, and perhaps thereby bridge another leap from the tangible to the inferential and imaginal, and back to the world of actions toward others.

            One cuts wood, carries water, and does math, about which more to be said elsewhere.

          • James Franklin

            Well put. Re ethics, you might like to consider my article 'On the parallel between mathematics and morals'. While empathy and compassion aren't mathematical, a lot about justice is based on equality (of persons).

          • G

            Hi James- Thanks! I just found your 'matns and morals' paper (Philosophy 79 2004), so I'm up for a good read as soon as I have time (this weekend).

            Agreed, empathy & compassion aren't mathematical. Also agreed, much about justice is based on equality. But there's something about the way you phrased that, which is tantalizingly suggestive of a relationship between justice and equality that is amenable to something like an empirical and rational treatment, in a way that might intersect some of the ideas I've been working with in recent years.

            Part of what I've been up to, is a fairly extensive 'ought from is' exercise, seeking to ground moral principles in empirical facts. Understood this is risky in that much criticism is leveled against any such attempt. But if one assumes that the issue of the existence of deities can't be resolved empirically (in essence, agnosticism, though I don't much like that word either), one needs to define a basis for one's moral system that is 'naturalistic' and not dependent on a firm answer (positive or negative) to issues of theology.

            I get the distinct impression that we are living in a time of much change in philosophy and in the belief systems that exist in our cultures and in global culture.

          • James Franklin

            I would say the main basis of morals is the worth of persons (and the equal worth of persons). In a way that's naturalistic (in that worth is inherent to persons and not imposed by gods) and in a way it's not (worth is not a scientific property, though it may supervene on naturalistic properties).

          • Jimmy

            "Clearly the objects of mathematics exist in nature: number, symmetry, and so on."
            Hmm, not so sure about that. You will struggle to find a lot of mathematical objects in nature - perfect straight lines, circles, infintesimals etc. They just aren't there, which if I remember my Plato right is one of the things he said.
            So are they in the mind of the mathematician ?
            I'm not so sure about that even. How do I know my imagination of a straight line is truly straight ? I can call it straight, sure, and I can write down a symbolic definition of straightness, but how am I going to measure my mental picture to see if it is truly straight, or look closer to it to see if there are any flaws ?
            To actually find a straight line in nature, or in the pictoral imagination is surely impossible.
            It may be that the ancients thought they could see straight things, but even then it must have been obvious that it's not really the case - unless they had very bad eyesight.

            But you say that our minds are a part of nature, and of course they are, so nature via the mind is developing all those mathematical relationships without showing us the actual objects that we claim to derive the relationships from.
            Where did the idea of straightness come from ?

          • James Franklin

            First, an imperfect circle or line is just as much a mathematical object as a perfect circle or line; and now with more sophisticated mathematical technology than the Greeks had, we can study imperfect circles etc too. The Platonists' purity/perfection fetish has been bad news for philosophy. Still, we have a concept of straight line (not the same as an imagination of it), which allows us to study lines, straight or near-straight

          • Jimmy

            Thanks for replying.
            Hmm, OK, my maths stopped at a-level, and nobody ever mentioned imperfect circles while I was doing it - it was either a circle or it wasn't. If post a-level study expounds on imperfect circles, whatever they may be, then so be it.
            Nevertheless I'd be interested to know if these imperfect circles have the same sort of relation to the natural sensory world as perfect cricles - ie, there are none to be found, truly, anywhere, whether physically or mentally.

            I'd also be interested to know what you say the difference is between a concept and an imagining of a straight line, and how you can have one without the other ?
            I confess I can't think about lines in any way without some sort of visualisation, especially after having studied, as anyone else, reams of representations from childhood on.
            If a crazed demon with a magic eraser managed to erase every single picture of a line and circle from all textbooks (and such), and drugged all people so that their imagination was blocked from visualising lines and circles - would the maths symbols still be comprehensible ?
            OK I may be deviating from the subject somewhat, but, still...

          • James Franklin

            Fair questions. A coin has a shape; an exact shape. It's the shape of an imperfect circle. That implies some mathematical facts about it, e.g. that if it doesn't deviate from the nearest perfect circle by more than 1%, its area doesn't deviate from the area of that circle by more than %.

            I agree that mathematical concepts generally require images. ("There is no thinking without an image", as Aristotle says). But that doesn't mean the concept just is the image; e.g. concepts have generality while images don't: the concept circle and its definition applies to circles of all sizes, whereas an image of a circle has some particular size (mental size, whatever that is ...).

          • Jimmy

            But, as there are no perfect circles, it's deviating from something that never existed except as a set of impossible rules. I'd be more inclined to think that geometric perfection is the deviation, really.

            I couldn't so easily say that I can hold the concept of a circle without an accompanying image, whether it's the usual classroom style textbook circle with undefined r, c, d etc.

            You could simply have the formula, written down, but that has no meaning without the accompanying visualisation unless you are claiming that the meaning is held in the symbols alone somehow ?

            It almost looks to me like there are two aspects of our brain, dealing with different domains of things, competing for our attention and being mixed up in a fudge of linguistic and pictoral soup that doesn't really work.

            But I know there are blind people who can do high level maths - no visualisation involved. There's a guy called Kent Cullers who has never had sight but he ran the SETI project for years - a skilled physicist.
            His understanding of geometry is (I can only speculate) mostly from the sense of touch, but I expect he experiences the same issues with the discrepency between formula and reality.

            Do we know how this sort of impossibly idealistic geometry arose in the first place ? It's a specialist subject that as a layman I don't see much of so I assume that somewhere in the ancient world, pre Euclid, someone decided to break with reality and apply impossibly perfect attributes to natural and variable objects.

          • James Franklin

            That's correct (and blind mathematicians are amazing). But you've got to think in terms of approximation, not idealization. Results about perfect circles (e.g. area formulas) apply to real wheels and coins because those approximate perfect circles. (Mathematical care is needed of course on what results do stand approximation, how good the approximation is etc.)

          • Jimmy

            Hmmm, but no quantity is actually really measurable in the physical and sensory world, no ? Once you let go of the sort of human scale you get all sorts of problems fluctuating boundaries due to the movement of atoms, ever increasing perimenters depending on the length of your measuring stick (after the style of coast measurements) - and that's not to go into whatever quantum strangeness there is. It's all very contingent and changeable, seems to me. So aren't the imperfect circles as elusive as the perfect ones, not least because whatever method used to measure the imperfection will inevitably utilise those perfect idealised objects, and be seen in relation to them ?

          • James Franklin

            If you want precision and measurability, there's plenty in physics - the charge on the electron, say, and all the electrons have it exactly. Plenty of exact symmetries too, in the laws.

          • Jimmy

            I did some googling and couldn't find wither way if we have a final definitive figure for the electron charge. I think it has been revised as instrument and experiment have been refined from Millikan onwards. Have they eliminated instrument error completely ?

          • James Franklin

            The issue is ontology, not epistemology: the electrons' having the exact same precise charge, not our knowledge or measurement of it.

          • Andreas Ramos

            Now, wait. The mass of the electron is known to eleven decimal places. But that is not the same thing as saying that the mass of the electron is precisely that. We only know to the limit of what we can test and measure. We will never know exactly the mass, so we can not say all electrons are identical. For our current level of measurements, yes, but that's our limit.

          • James Franklin

            It's a matter of inference to the best explanation. The most likely reason that all electrons have the same charge as measured to 11 decimal places is that they have the same charge exactly. Cf Euler's inference that the sum of the squared reciprocals equals pi squared over 6 to several decimal places, which is most likely because it equals that exactly.

          • Andreas Ramos

            The key phrase in that reply is "most likely". But we can't say "it is precisely". We can only know to the limit of our measurement. The physical world is not the same as the ideal world or world of ideas.

            The more this discussion continues, the more I think mathematics is a branch of theology, not science. Mathematics has an remarkable ability to describe the physical world, but it is not identical to the physical world. Math is not physics.

            If it were, we would only need chalk and a blackboard to discern the nature of the world. There would be no need for messy experiments. Mathematicians could tell us all there is to know. They wouldn't even call it predications: they would know with certainty. No need to test.

            But it doesn't work that way. The world has an annoying way of rebelling against our beautiful ideas.

          • James Franklin

            Math is not physics, as you rightly say. That's why physics needs experiments and mathematics just the chalk and blackboard, which discerns the (mathematical, not physical) nature of the world.

          • Andreas Ramos
          • Ye Olde Statistician

            Thinking in images is the imagination; but we also go beyond them using intellect, by which we reflect upon the images and abstract "concepts" from them. It is this transition from percepts to concepts by which we pull the perfect circles of the intellect from the imperfect circles of the senses. And these perfections provide us with reasonably precise mathematical models of the imperfect physical world.

          • James Franklin

            That's correct. (almost). As explained in the book (An Aristotelian Realist Philosophy of Mathematics), we can sense basic mathematical properties but on top of that there's an intellect that's more conceptual and can manage concepts like "perfect circle". I don't know about "imperfect": it's not as if coins *ought* to be circles, they just have a near-circle shape, that is as much a real shape as a perfect circle and can be studied by geometry.

          • Andreas Ramos

            Would that be a crazed Cartesian demon?

          • Jimmy

            It moves way to fast for me to see.

          • Andreas Ramos

            That's an interesting idea. How do we know if our thoughts are accurate? How would we test our concept of a straight line? Some may say it's straight by definition, but there are plenty of ideas-true-by-definition that turned out to be false or useless (the christian god, for one).

          • Jimmy

            I've put this to a few people and many don't get the question, even though I'm sure it's a very old and basic one going back to Plato and beyond. I think this is probably due to habit and training. I went through the usual schooling and pre college maths/physics training and never once was the question raised - rather we were simply told to "imagine a straight line A to B", or "imagine and infinite line stetching", and I guess it was assumed to be impractical to do anything other than go along with it. For my experience questions like that are really left for the classes on Plato's Republic, if you are doing a philosophy course, and you can have a bit of fun catching people out. Prof Franklin seems to be well versed in all this.

          • bobthechef

            How do you know testing is a sensible idea? Even Pyrrho had the good sense not to pursue his skepticism aggressively. It's self-refuting.

          • HaakonKL

            Proof sounds like a good start.

            I don't really see how pontificating about egg-arrangements prove that math is not rational instead of empirical?

            If the world did not exist, would not all of mathematics still hold? Can not mathematics describe worlds that in no way can exist?

            You have shown that you can map between the realm of mathematics and our realm, but you have not shown them to be the same, have you?

          • James Franklin

            If the world did not exist, all mathematics would still be true (of possible worlds). Worlds that can "in no way exist" could be hard to describe. "Square circle"? That's not description, it's putting incompatible words together and failing to describe.

            You can only "map between" domains that have structure in common.

            Use of "pontificating" about someone's argument says more about the user than what's attacked.

      • Ye Olde Statistician

        It seems to me the article was a nice illustration of the Aristotelian dictum that "nothing is in the intellect that was not first in the senses" but that human understanding does not end with the senses.

        • James Franklin

          That is right. This philosophy of mathematics is not called Aristotelian for nothing.

    • John Robertson

      Oh, my. Charles Sanders Peirce, who some believe is the greatest mind ever produced on American soil, in the 19th and early 20th Century advanced Aristotelian thinking (realism) far beyond the poor nominalistic thinking of modernity. Why not try him to help understand why Aristotle was spot on?

      • James Franklin

        That's correct. Peirce's thinking was quite close to Aristotle's scholastic followers. And he had a good appreciation of mathematics and probability. But on the issues strictly of the philosophy of mathematics, in my opinion not quite realist enough.

        • John Robertson

          Hmmm. I would like to plumb your mind to understand why the philosophy of mathematics escapes the purview Peircean semiotics, his broader study of signs--which seems to transcend his 19th century approach to mathematics and probability. Maybe I don't quite understand what your definition of "real" is.

          • John Robertson

            I would be hard put to believe that "Peirce is not quite realist enough!"

          • James Franklin

            Mathematics the human activity doesn't escape semiotics, but I was concentrating on what mathematics is about, real-world properties like symmetry. "Real" in that sense is basic and surely not definable in terms of anything simpler.

        • Nick Maley

          Great article, James. But John Robertson is correct about . Peirce. He had an extremely sophisticated semiotic theory, one that was realist to the core. In Peirce, representation is always grounded in real objects. There must be a connection between Peirce's semiotics and his quasi Aristotlean realism about Mathematics.

    • James Franklin

      Well no. Relationships are as real as properties (despite a long Western history of regarding them as not quite real). If mathematics is about relationships, it's about something real.

      • Andreas Ramos

        James, that's precisely the problem. You declare relationships to be real, which you use to declare mathematics is real, and you then declare the world is mathematics and conclude there is only mathematics.

        All of this depends on the assignment of "being" to relationships.

        You add "(despite a long Western history of regarding them as not quite real)". Actually, it's the opposite. Greek philosophy and Catholic thought has regarded ideas as real. It's only in the last three hundred years that scientists have rejected this (but the majority of Americans continue to believe in angels and gods).

        • Ye Olde Statistician

          Greek philosophy and Catholic thought has regarded ideas as real. It's
          only in the last three hundred years that scientists have rejected this

          You mean to say that scientists haven't had any real ideas since then?

  • Tom Hickey

    You do realize that in favoring Aristotelian realism you are either agreeing with Aristotle's epistemological explanation in terms of intellectual intuition of essences and causes, which modern science rejects as without basis, or you are relying on a new epistemological theory of realism based on scientific evidence in cognitive psychology, I would presume. Which is it?

    • James Franklin

      The answer is: some of each. For the perception of symmetry, numerosity etc that we share with animals, you just need ordinary scientific perceptual psychology (whatever the correct philosophy of that is.) For the higher levels of mathematics like proof, visualization and understanding, you do need an intellectual intuition of - well, not exactly essences and causes, since you don't have those in the category of quantity, but certainly intuition of necessary interrelationships; as in the 2x3 =3x2 example in the article. Modern science surely doesn't reject such interrelationships; on the contrary, it relies on them to study almost everthing.

      • Tom Hickey

        It may rely on them, but science doesn't trust intuition without testing it. That's the difference between science and philosophy. Assumptions are sometimes based on inductive reasoning and at other times abductive. Some call abduction "intuition," and others call it "guessing." At any rate, what is generated from assumptions is hypotheses. These hypotheses get their logical necessity as theorems by postulating the assumptions as axioms. I don't know any scientists that think that hypotheses as empirical protocols are any more than highly probable however. That is to say, science is tentative on future discovery rather than dogmatic about assumptions regardless of how intuitively they may be held.

        The dilemma I am proposing with respect to mathematics and logic is that either human have intellectual intuition, in which case that must be explained cognitively and hasn't, or math and logic are constructs that happen to be useful in dealing with the world but we cannot explain how that correspondence arises.

        I simply don't buy the "realism" claims other than as naive realism. This issue has been debated at least since the ancient Greeks, in the West at least, and remains unresolved. So I think that those holding a realistic position have to answer Hume. Is it knowledge or belief and if knowledge, how so? There have been many answers given by philosophers but none has carried the day, and cognitive science is in its infancy with respect to resolving the age-old epistemological questions.

        I bring this up because you began with ancient philosophy and seem to align mathematical intuition with Aristotle's realism. I don't see Aristotle has being helpful in this, first, because almost no one today other than Thomists accepts Aristotelian epistemology, and secondly, since scientist view Aristotelianism as having reinforced the dogmatic approach that delayed the scientific revolution in the West for centuries if not a millennium.

        If the argument were as simple as you make it out to be, there would not have been the long history of intellectual debate over theory of knowledge that is as yet unresolved, either by reasoning alone, which is philosophy, or theory for which the evidence speaks, which is science. Math cannot decide this, since it concerns philosophy of math and there is ongoing debate over this, too. The jury is still out, e.g., between the intuitionists and the constructionists.

        • James Franklin

          I vote for "human have intellectual intuition, in which case that must be explained cognitively and hasn't". It's true that cognitive events like understanding why 2x3 must equal 3x2 are a problem for many epistemologies, so there's "ongoing debate". There's no future in denying the evidence though, and accepting simple epistemologies into which the facts don't fit.

          • Tom Hickey

            Good luck arguing that with constructivists. You are simply begging the question if you can't explain what you claim, but rather just assert it as self-evident when many disagree that it is self-evident other than in a particular POV that is attempting to assert itself as universal and absolute. It doesn't move the debate forward since it is not a new claim.

            This is the position of naive realism, not Aristotle, who was not naive about the debate or the issues. What is new is more people in a variety of fields are now trying to resurrect "Aristotelian realism" without understanding what Aristotelian realism actually is and are just redefining it, using Aristotle as an argument from authority. I would advise just dropping the reference to Aristotle unless you make use of it in explaining intellectual intuition.

            I am not pushing any view here. Just sayin' that it’s explanation that is at issue. I happen to like the intuitionist perspective, but I also admit that it stands on weaker explanatory ground than the constructivist at this juncture. I would like to see it advanced, but I don't see it so far in what you have said.

          • James Franklin

            There's no point in arguing with constructivists, since they don't accept the objectivity of argument. Intuitions like why 2x3 = 3x2 are self-justifying (a very Aristotelian view); theories of epistemology have to account for that, not photoshop intuitions out of the picture in favour of some simplistic "naturalist" view.

  • PhilMathGeek

    I'd be interested to know whether the author's Aristotelian realism admits transfinite arithmetic, or set theory, or other elements of mathematics whose physical realization appears, if not impossible, at least non-actual.

    • James Franklin

      The answer is yes, in the same sense as the science of colour admits a shade of blue that may happen not to be instantiated. It may be that the universe is finite, but infinities are capable of being realised. It's the business of mathematics to study realisable properties, not to decide which of them happen to have been realised by the course of nature (or the will of God, or whatever decides these things).

      • James Franklin

        Having said that, I agree that uninstantiated universals are the most serious technical problem for an Aristotelian realist philosophy of mathematics. Chapter 2 of the book (An Aristotelian Realist Philosophy of Mathematics) deals with the issue.

        • bobthechef

          I think we must distinguish between class and set here. I can comfortably talk about the class of apple trees, but sets permit us to refer to say the set of "these three arbitrarily chosen trees" which we can discern as a purely mental distinction and not one with any real existence outside of the mind.

          • James Franklin

            There is a difference between the two cases but ... the heap of those three trees is there and consists of three trees (prior to any mental act or "arbitrary choosing" of ours), so that set is not a purely mental distinction. Still, the class of apple trees is more natural, being determined by a natural property, being an apple tree.

    • bobthechef

      You might appreciate Lesniewski's response to set theory (mereology, which he grounded in two other theories, ontology and protothetic). Lesniewski was a nominalist as far as universals are concerned. To him, the class of philosophers was a real, concrete object in the world; if only one philosopher were to exist, then he would be that class; if no philosophers were to exist, then the name would refer to nothing.

      Oddly enough, and this may be consistent with Aristotelian thinking, Lesniewski believed that set theory had been poorly axiomatized and betrayed Cantor's and everyday intuitions about sets (he saw ZFC as a hack to get around Russell's paradox, a paradox which cannot occur in mereology).

  • SaintMarx

    Your first example is that of "the motion of a cat thrown out of a window"? This attempt at humor expresses psychosis. I stopped reading at this point.

    • PvNP

      The "cat out of a window" refers to a typical problem/example used in physics classes where they found out that cats that fell out of windows at a certain height did not survive the fall, but cats falling out of higher windows did (this was studied by looking at reports of cats that had fallen out of windows, not by researchers actually tossing them out). The author is clearly referencing this well known odd result (hence the dichotomy between physics--what motion was the cat taking? with physiology--why did the cat going faster survive?). (The difference, by the way, is that if the cat was thrown from a great enough height that it achieved cat terminal velocity, then it lost the sensation of falling and relaxed its muscles which gave it a much better chance at survival). You can do the physics computation and find that terminal velocity is hit at something like 7 stories, so a cat dropped from the 5th floor will not survive, but one from the 7th will.

      Anyway, it is time for everyone on the internet, like @SaintMarx:disqus to take a big collective deep breath and stop taking offense over every tiny thing. Geeze. If you don't want to read an article, don't read it, but don't get up on some high horse and tell all of us about it.

    • M. Ratcha

      Funny. I kept reading, but I did immediately take note. If the cat-window example is a physics inside-joke, it reveals a lot about physicists (and the mathematicians and math philosophers who promulgate this kind of inside joke). Elsewhere, I've been reading a great deal of hand-wringing material by scientists and "science communicators" who are worried that the general public is disenchanted with science. Our scientists and mathematicians are the people who throw cats out of windows in the name of their pursuit of truth (capital-T Truth? quote-unquote "truth"?) and then think it's ha-ha funny to reference the act.

      No wonder there are so many nominalists. If we can view math as a language spoken by cat killers---say, lofty mathematicians and morally bereft scientists and engineers who speak only in numbers---it's easier to turn away from it. Unfortunately, this means it's easier to turn away from science altogether. People speaking their own internal/private language throw cats off of buildings and speak arrogantly to and of the rest of the population... which then doesn't know whether to trust a damned thing any of Those Guys say (and oh my, it's largely guys, isn't it?). A digression from the subject at hand? Perhaps, but it offers perspective that does, indeed, relate to the real world, beyond its egg boxes.

  • johnmerryman

    When we think, we have to distill enormous input into relatively concrete impressions and thoughts. Think in terms of a camera taking a picture. Shutter speed, aperture setting, lighting focus, etc, or it's all blurred and whited out. So we naturally lock onto those patterns which are most ordered and structured as a shortcut to understanding the world around us.
    The psychology of math is every bit as important as its logic. What is more basic than the point, line, plane and area of geometry, yet according to simple multiplication, anything with a zero dimension doesn't exist. An infinite number of dimensionless points don't really add up to a line, anymore than any multiple of zero adds up to anything more than zero. Yet it becomes too intellectually complex to keep insisting these points, lines and planes have some miniscule dimension in order to exist, so we overlook the multiple of zero issue.
    Then again we view space as fundamentally three dimensional, since that is the system by which we define it, but if we don't specify the actual coordinates being used, this system is meaningless, yet if we do specify them, then other coordinate systems can equally be used to define the same space. Just ask the Israelis and Palestinians about using different coordinate systems to define the same space. So, effectively, space is infinitely dimensional, but our mental processes much prefer the short version and the ecology and evolution of belief tends to enforce our preferences.
    Math is a form of information and information is inherently static, yet it requires energy to be manifest, but energy is inherently dynamic. There is no information in a void. Mass is energy in a state of equilibrium and thus marginally static, so it is not as though it is just a property of our imagination. In fact, over the course of billions of years, we have developed a central nervous system to process information and the respiratory, digestive and circulatory systems to process energy. So we try to discretely comprehend what insists on flowing continuously.
    Math is order and our sense of order still depends on our point of view.

  • Earthstar

    So, you managed to write on mathematics, philosophy, reality and Aristotle without once mentioning or touching on geometry? That's an accomplishment.

    And strange.

    • James Franklin

      A good point, but geometry is tricky when it comes to philosophy. Is it provable mathematics as Euclid thought, or is it physics about the shape that space happens to have, as suggested by the discovery of non-Euclidean geometries? I promise a chapter on geometry in An Aristotelian Realist Philosophy of Mathematics.

  • gogododo

    So maybe math exist in an absolute since, but to process mathematical
    reality we first need to filter it through our consciousness, which I
    take as a indicter that math is part of our self awareness. Maybe
    it also exist in an absolute since as the author implies. There is no
    knowledge i'm aware of that disputes this. But at the same time all
    evidence that we know of, in supports of reality having a numeric
    skeleton is itself filtered through human conscious. Maybe plants or
    mushrooms are aware of math but we don't know what they are aware of so
    it's a moot point. So to me it will remain an unknowable question. Of
    course you could say the consciousness in which we are filtering a math
    engorge reality is itself part of the math engorged reality...

    Oh what a fun rabbit hole to find one selve in.

    • Tom Hickey

      This was essentially the view of Plato and Eastern thought. I am partial to this view since I think it is the only way to get past the mind-world dichotomy that has dominated debate in the West since Descartes.

      Dualism makes realism pretty much impossible to establish other than as a belief, since there needs to be a link established between mind and matter. There cannot be a third thing that acts as like, since this involves an infinite regress of further links.

      More promising is denial of dualism and assertion of monism. They leaves monism of matter, reducing mind to matter, or idealism, incorporating matter into mind.

      Or positing consciousness as the basis of mind and matter with the structure of consciousness reflected in both mind and matter. I think this is most promising but it still lack explanation.

      Plato spoke metaphorically about this, and this is not considered to be an explanation today. In short, I suspect that this issue is a subset of the problem of consciousness, for which as yet no theory has been advanced that is either logically compelling or warranted by evidence scientifically.

      This is actually close to the Aristotelian - Thomistic view of being as intelligible as well as intelligent. Humans know the intelligible aspect of being directly through intellectual intuition as explained by Aristotle in terms of the active intellect impressing the forms of things on the passive intellect. Intelligible being and intelligent consciousness are united in Aristotle's conception of God as self-knowing knowingness as the identity of being and consciousness.

      This identity was also the basis of Plato's thought but he held that knowledge occurs within consciousness directly when material objects present themselves, reminding the soul of what it already knows.

      This is also the view of the ultimate principle in various expressions of Eastern thought. In the Vedic tradition, there is a well-developed theory of the identity of name and form in unitary consciousness, which can also be seen as underlying the Platonic account. See E. J. Urwick, The Platonic Quest (London, 1920), for instance.

      Anyway, this is all speculation, and at this point we don't know how it works in a way that passes the tests of modern knowledge.

  • Just Wondering

    "don’t get blinded by shuffling symbols, don’t disappear into a realm of abstractions, just keep an eye fixed on the mathematical structure of the real world."

    Weren't Complex numbers (sqrt(-1)) discovered using mathematical abstractions and later applied to physical reality?

    • James Franklin

      It did look like that at first, but with the discovery of the Argand plane and its transformations, the connection of complex numbers and reality was understood.

  • Andreas Ramos

    The flaw in James Franklin's article is unclear thinking. There are points in the paragraphs, but you have to unravel the sentences and ideas to understand what he's trying to say. I learned long ago that when someone writes like that, it's to cover up poor thinking.

    I'll give you an example and show you how his overly-complex thinking messes up the issue. It's not his habit of flinging cats out of windows. It's the eggs.

    Franklin argues eggs are sold in twelves because eleven is a prime and twelve can be arranged neatly in several ways. Having solved that puzzle, he moves on.

    Well, no. Why are eggs sold in twelves? Franklin hasn't been to China, where eggs are sold in groups of ten. "Dozen" is a Western cultural artifact. Franklin also hasn't been to Europe, where at a farmer's market, you can buy eggs in whatever number you want: three, five, eleven, whatever. Or just one. Why does Franklin think the arrangement of eggs is a feature of the universe? Because he thinks eggs come in an egg carton at the supermarket. Egg cartons (which Brits call "egg boxes") are only a hundred years old. Before that, eggs were carried in small baskets, which didn't need mathematical ordering nor had anything to do with primes.

    See? He takes small commonplace experiences and spin off into irrelevant complexities. The rest of his article is the same.

    Mathematics isn't something that only a professor of mathematics can understand: infants and crows can count. I suspect the nature of mathematics can also be explained in clear and simple language.

    • James Franklin

      Egg cartons are a cultural construct, or invention. Like other pieces of engineering, their success or otherwise depends on objective, non-constructive facts, such as that 10 and 12 are not primes.

      • Andreas Ramos

        You miss the point. Egg baskets are also "... a cultural construct, or invention. Like other pieces of engineering, their success or otherwise depends on"... well, it's irrelevant to egg baskets if the number of eggs are prime or not. You're arguing backwards to make your point.

        • James Franklin

          The picture at the head of the article explains better than any words could why primeness is relevant to egg baskets. What's that hole doing?

          • Andreas Ramos

            That photo is not an egg basket. That's an egg carton. If it were an egg basket, your example would be pointless: primes are irrelevant to egg baskets. An egg basket can easily hold 3, 5, or 7 eggs.

            The missing egg makes me wonder about how you approach mathematics. You're assuming ideals of math and then applying them to the world. You look for things that match and use that as proof. LIke the egg carton, that's a backwards proof.

            Others talk about axioms, and that reminds me of the Ontological Proof for the existence of God. If you've studied philosophy, then the origins of math and Christian theology are intertwined. For many of the Greek philosophers, mathematics was occult. Which makes me wonder if modern math is a sort of theology without personification.

            For example, look into Pythagoras. Not just the theorem; look into what Pythagoreans meant for the Greeks. It veers rapidly into astrology, prophecy, and other things. All of that has been brushed away (okay, buried) in modern mathematics, but it's still there in the framework: using mathematics to discover the unseen, thinking that ideal mathematics is more real than fallen things, and so on.

            Is mathematics a science? I don't mean if it's like biology, where you go out and discover a previously-unknown bug. Is it a science like theorems in physics? The modern scientific method includes falsifiability, and that doesn't exist in math, does it? It will never happen that 2+3=5 can be shown to be false. But what if it could? Now that would be interesting.

        • Ed Lake

          You seem to have got really stuck on the eggs thing. Franklin chose egg boxes as a fairly commonplace example of a rectangular array. Of course you don't *have* to arrange eggs in a rectangular array, but it *is* blindingly obvious when a collection of eggs is so arranged. Which is the point.

        • James McMullen

          The flaw in your comment is unclear thinking. Eggs sold in groups of 10 illustrate James Franklin's point just as well as those sold by the dozen. If you buy three, five or eleven eggs you will be unable to factor them into neat groups because those numbers are prime. That is in fact a feature of the universe. He's not really writing about eggs: he's actually writing about mathematical abstraction. The egg baskets are cultural constructs, true, but this is not remotely related to the article.

          • Andreas Ramos

            James: Franklin wants to argue that eggs can't be sold in primes because they can't be factored neatly. But that's true only with egg cartons. If you buy eggs with an egg basket, then factoring is irrelevant. This means there's no point to the egg carton example. He just wants to say that primes can't be factored and then hastily looks around for a convenient example.

  • atimoshenko

    pure mathematics reveals to us the topography of a region whose truths pre-existed our investigations and even our language

    Why is this a problem? Surely any model should reflect the thing that it models over the region in which the assumptions chosen for that model hold? Chose your set and operation – define your group.

    • James Franklin

      "Choosing", "defining", "postulating", "imagining", "hypothesising" or any other mental acts won't have any effect whatever on whether the primes thin out logarithmically. That's an objective fact about how numbers are.

  • foobie

    Sorry, but you're going to have to try harder if you are going to argue against a nominalist understanding of mathematics (to which your comic-book description does not anyway do justice - nominalist foundations have no problem with, e.g., differential geometry) - it is not obvious to me that - say - ultrafilters, non-standard models of ZFC, the continuum hypothesis, or non-measurable sets (to think of a few mathematical concepts that have no direct appeal to intuition) have any convincing ontological status.

    • James Franklin

      It's not obvious to me either. That doesn't bear on the obvious ontological status of simple arithmetic or say discrete optimization, or the objectivity and perceivability of the ratio of your height to mine. As to "trying harder", you can see how I tried harder in the book, An Aristotelian Realist Philosophy of Mathematics.

      • G

        You're on my reading list now.

        Re. intuition, I would say that it's a useful source of raw materials that need to be further refined via empirical and logical means. Neither is better than the other; the hammer is for nailing beams, the trowel is for smoothing concrete.

  • Jon

    Now wasn't that dogmatic.
    Someone explain to me how he showed that the impossibly rich abstracta of pure maths that can't be observed or derived directly from the world doesn't either belong to another "realm" (platonic) or is simply the result of extrapolating from rules? There are obvious problems with the others, but his was by far the least convincing

  • The Art Guys

    Symmetry is not necessarily a mathematical property.

    • James Franklin

      Why not? E.g. what kind of symmetry would you suggest that isn't a mathematical property?

      • FA Miniter

        All denumerable infinite number sets have symmetry, for instance. Though, nondenumerable sets are nondenumerable precisely because no symmetry can be established.

        Particle physics deals in symmetries at all levels, from the structure of classifications of leptons, quarks and baryons, to the fact that the creation of a lepton or quark requires also the creation of an anti-lepton or anti-quark. And the destruction of one requires the destruction of the other.

        Curiously though, it seems that conditions existed in the early universe whereby C, P and CP parity violations could occur, thus resulting in a universe with something rather than nothing.

        • James Franklin

          It's because of all that symmetry in physics that Herman Weyl introduced so much group theory into physics, to study it.

  • FA Miniter

    While it is true that mathematical concepts are not "abstracta [that] float somewhere outside our own universe of space and time", as the article would have us interpret Plato, it is also true that mathematical concepts exist independently of the Aristotelean realism that the author insists upon. The concept of π is completely independent of any actual circle, but is a necessary part of the concept of circle, for instance. Mathematical concepts are perceived simply because they form part of the structure of the universe. Avogadro's Number may provide us with the border between quantum mechanics and general relativity. Numbers are not outside the universe; they are not just within it. They are part of space-time, inextricable from that which gives the universe form.

    • James Franklin

      π is realised in every circle, as the ratio of its circumference to its diameter, so it's "in" the universe in that sense. Any independence it has is just like e.g. the fact that orange is between red and yellow: it's just how it's got to be. (Also, I didn't "interpret Plato": Platonism as understood in philosophy of mathematics is a simple theory that's not true to the real Plato.)

  • jaycal33

    As the author is a professor of mathematics, his views merit serious consideration. He writes "Here’s one: if abstracta float somewhere outside our own universe of
    space and time, it’s hard to imagine how can we see them or have any
    other perceptual contact with them. So how do we know they’re there?".

    In my opinion, mathematics creates a world, a universe based on axioms. It is a world that needs no physical basis, it can just be based on thought. From the axioms we go on to derive results (lemmas, theorems etc.).

    If the behavior of anything in the physical universe can actually be explained/predicted by mathematics, well and good. However, mathematics itself does not need the physical universe. All it needs is that the its theorems do not violate the axioms.

    One can quite easily think of axioms and resulting theorems that have no relation to anything at all in the physical universe.

    Of course, one can ask "what is so sacrosanct about not violating axiom?". The answer is that mathematics of that sort would not be an intellectual challenge, nor would it be useful in explaining anything in the physical universe. Anything and everything could be "proven". Lack of rigor robs mathematics of its power.

    Mathematics is precise thinking, the building up of theorems consistent with the axioms. It is good training for the mind.

    There are no doubt yet unthought of axioms and theorems based on those axioms. To answer the author, we will know they "existed" when we discover them. What we do know is that we don't know every possible axiom and every possible theorem that will follow?

    Did set theory "exist" before Cantor wrote his paper in 1874? A mathematician may ask, what does "exist" mean, can you give me an axiom?

    • James Franklin

      Real mathematics is not done by dreaming up axioms and deriving theorems from them. Mathematics is more serious than that. You can invent any number of axioms for things something like numbers, but 2x3=3x2 is just true - not true in some system, or based on thought, or ... just true of numbers and of discrete arrays of real-world objects.
      Sets before Cantor ... "Now Abel kept flocks" (Genesis 4:2). Flocks are sets; you count them. If you want theory, there was plenty of combinatorics long before Cantor, which is about counting large structured sets.

      • jaycal33

        Prof. Franklin, you have to explain the difference between "mathematics", "real mathematics", "false mathematics" etc.

        Hilbert is reported to have said elements, such as point, line, plane, and others, could be substituted by tables, chairs, glasses of beer and other such objects. I believe that Hilbert was saying that mathematics could be applied equally to different physical objects, or even to no physical objects at all. I believe Hilbert's point was that mathematics can be divorced from its applications to the physical world.

        I don't understand why you think mathematics cannot be done by "dreaming up axioms and deriving theorems from them". If the axioms and theorems are sufficiently non-trivial they should interest a mathematician, especially one who is tenured and doesn't have worry about publishing! Maybe a century later there will be some great application found for that mathematics.

        Is the standard you set for mathematics your personal standard, or is there some objective basis for it?

        Human knowledge also advances by also discovering knowledge that doesn't immediately relate to the world as we know it. Do you believe it is impossible to come up with a set of axioms and resulting theorems that are non-trivial and yet do not relate to the physical world as we know it, and if you do then why so?

        • James Franklin

          Can you name then any example of playing with arbitrary axioms that then turned into accepted good mathematics? I can't recall any. There have been some debates about putative cases, such as non-Archimedean analysis (analysis without the Archimedean axiom). It didn't make much impression and other mathematicians suspected it was just a wank. If it had turned out to describe something, that would have been convincing, but in the absence of that, no.

          There are instances where topics of initially only pure mathematical interest turned out later to have applications (e.g. factorization with large primes in crytography), but that's quite different from playing with arbitrary axioms.

          • jaycal33

            Probably there is still enough mathematics that relates to real world left to be discovered. However, there is no reason to say that something that is not related to the real world may not be developed in the future.

            The real world is of course a great inspiration for mathematics, for example if one thinks about speeds, volumes etc., then calculus naturally follows. An exceptionally creative mind may not need any inspiration from the real world to think up of axioms and theorems.

            In Arthur C. Clarke's "The City and the Stars", humans in the distant future spend their time re-discovering mathematics that does exist in their knowledge base, to experience the pleasure of discovery. Mathematics is 4,000 years old. In 40,000 years we may have run out of axioms and theorems that relate to the real world. To keep obtaining the pleasure of discovery, mankind may turn to formulating non-trivial axioms and theorems that do not relate to the real world. Alternatively, they could spend their time re-discovering something already discovered as they do in "The City and the Stars".

            At this point the issue between us maybe merely semantics.

            I may define mathematics to be "non-trivial axioms and resulting theorems".

            You may define mathematics to be "non-trivial axioms and resulting theorems that have applications (relate to something in the real world)"

            I doubt that it can be proven that there are no non-trivial axioms and theorems that have no relation to the real world, even though there currently may not be any good examples. I am sure discovering non-trivial axioms without the inspiration we get from the real world is much harder.

            We can both define mathematics in different ways, and neither would be wrong I think.

  • Pete

    Great article on the philosophy of mathematics.

    I think that some form of realism, whether it be of the Platonist or Aristotelian variety, is a no brainer when it comes to mathematics. Some worry about the idea that these "abstract" mathematical structures can actually exist. Funny thing is, those same individuals haven't quite examined what our current scientific understanding says about many 'material' objects that they are quite happy to say exist. Physicalism, a doctrine that virtually all scientists and a large portion of philosophers subscribe to, starts to run into problems on closer examination. I have always considered myself a physicalist as well, but after really contemplating what that means I start to revert to simply considering myself a naturalist, which contrary to many opinions can embrace abstract objects without a problem. When one really begins to delve into the nature of the "real" objects that are out there, their decidedly ephemeral nature is exposed. For one, objects in the universe are made almost entirely of empty space. The atoms that compose physical entities are something like 99.999% empty space, with a very tiny nucleus and some extremely small electrons that move in very spooky "probability clouds" around it. The nucleus itself is composed of even tinier quarks and gluons, the true nature of which (along with every other elementary particle in existence) is extremely hard to pin down. Maybe they're "vibrating strands of energy" as String Theory posits, or “knots” in the fabric of space-time as postulated by Loop Quantum Gravity, whatever any of that physically means.

    The point is, what we think about as being physical is really no such thing at all. At the bottom, it really seems to become mathematical equations and relations. And at that point, things like the "no miracles" argument and the idea that mathematics is indispensable to our understanding of the world, you have a strong case for mathematical realism. The ironic part of the opposite position, that of nominalism with regard to mathematics, where something must be physically instantiated in space-time if it’s to be considered real (something which makes mathematics nonexistent), is its inherent assumption of physicality that is largely misunderstood and nonexistent in the first place.

    • James Franklin

      If you think physical things are "knots in the fabric of space-time", you've still got the physical reality of space. Anyway, if you accept realism about mathematical objects, at least you don't have to put up with them being 99.999% empty space ...

      • Pete

        Very true, but then what is this 'space-time' made of? I feel like its mathematics at the foundation any way you look at it. We're on the same page; its just the details of the particular brand of mathematical realism that we have to attempt to figure out. More articles like this are needed

  • Paul Braterman

    I can see how Aristotelian realism applies to properties like the symmetry of an egg, whose rotations have the same structure as a particular point group. I don't see how it helps us understand the status of statements like "Pi is a transcendental number", to which there is no corresponding physical fact.

    • James Franklin

      There are certainly physical facts corresponding to the irrationality of pi, e.g. about the failure of certain rotations of the circle to ever return to their starting point. However, I wouldn't say that all of the complexities of pure mathematics, such as transcendentality, mean a lot in terms of the physical world. It's rather that the basic objects of mathematics are or might be literally realised in the physical (and any other) world; pure mathematical research into those properties themselves can reveal a lot of facts that might be hard to interpret physically. Cf with colours: the complex betweenness relations in colour space are about the colours themselves; who knows if those are visible physically?

      • Paul Braterman

        I think there are problems here even with the irrational,
        let alone the transcendental. If real space is not known to be (let alone, is
        known not to be) perfectly Euclidean, and/or if distances are not definable
        with perfect precision, I see no way of distinguishing, by observations on circles, between true pi and a sufficiently long but finite decimal
        approximation to pi, even in principle.

        • James Franklin

          I can't let you slip from ontology to epistemology like that. If physical reality is exactly Euclidean (which indeed we don't know), then the irrationality of pi is realisable physically. Our "inability to distinguish" is an irrelevant matter of epistemology.

          • Paul Braterman

            Fair point. In some kind of possible physical world, at any
            rate, we could with sufficiently good equipment demonstrate the inadequacy of any rational approximation to pi. Perhaps you could deal with transcendence in the same way, by demonstrating the failure of any root of an equation to return
            the circle exactly to its initial position.

            How distinct are epistemology and ontology? I am thinking,
            rather obviously, of the Copenhagen interpretation of quantum mechanics, which seems to slide from “cannot be
            observed even in principle” to “there is no such thing as...”

          • James Franklin

            The Copenhagen interpretation's philosophical naivety on this point is a main reason for its decline.

          • Paul Braterman

            I naively taught the Copenhagen interpretation to chemistry undergrads for many years. What should take its place for this purpose?

          • James Franklin

            They haven't sorted out what the right interpretation is. I'm no expert. The Wikipedia article 'Interpretations of quantum mechanics' is informative.

  • 013090

    This is a very interesting article; I enjoyed reading it. I can understand how certain mathematical concepts are 'emergent' in the Aristotelian sense, such as ratio and symmetry. But what of the mathematics behind the laws of physics, the very foundation of our reality? How can one argue that they are emergent, rather than essential in a Platonic sense? How is there order rather than disorder? Would an Aristotelian say that order and disorder are just emergent concepts?

    • James Franklin

      I wouldn't call ratio and symmetry "emergent", and I think that word is too confusing in the context. The universe has some mathematical structure, e.g. that described by the symmetries of General Relativity. I don't see anything Platonic about that.

  • Roy Niles

    Mathematics is a measuring tool. It cannot plan or strategize. Strategies for example lay out such things as egg cartons, not mathematics. Strategies are the conceptual basis of intelligent activity. Mathematicians seem to think they've used mathematics to conceive of strategies when in fact it's the reverse that happens.

    • James Franklin

      Strategies, like any engineering, need to conform to the properties of what's being organized, just as sculpture needs to conform to the properties of the marble. Take Euler's famous example of the bridges of Koenigsberg. You can dream up whatever strategies you like to walk across the seven bridges, but mathematics tells you you can't walk across all the bridges without walking across one of them twice.

      • Roy Niles

        That's a rather pointless answer, since without the strategies that tell you how and why to build the bridges, including how humans have been devised to walk, crossing any twice would never need to be considered.

      • Roy Niles

        That's a rather pointless answer, since without the strategies that tell you how and why to build the bridges, including how humans have been devised to walk, crossing any twice would never need to be considered.

      • Roy Niles

        That's a rather pointless answer, since without the strategies that tell you how and why to build the bridges, including how humans have been devised to walk, crossing any twice would never need to be considered.

      • Roy Niles

        That's a rather pointless answer, since without the strategies that tell you how and why to build the bridges, including how humans have been devised to walk, crossing any twice would never need to be considered.

  • Mike Archbold

    Hegel built mathematics into his ontology. You don't see it discussed much, but it is a part of his triadic system.

  • Commander Howdy

    I enjoyed this article, another great few moments at Aeon, thanks!

    I wonder whether Maths is a great a wonderful intellectual fuel, allowing us to power other vehicles for understanding the world around us; not in and of itself a reality or containing the power to prove understandings of reality but a method, and a great one at that. I guess I line up with the Skeptic Postmodernist's mentioned in the article in that regard, a conclusion I find troubling and yet…there's always Faith.

    Quote: Boscovich;

    But if some mind very different from ours were to look upon some property of some curved line as we do on the evenness of a straight line, he would not recognize as such the evenness of a straight line; nor would he arrange the elements of his geometry according to that very different system, and would investigate quite other relationships as I have suggested in my notes.
    We fashion our geometry on the properties of a straight line because that seems to us to be the simplest of all. But really all lines that are continuous and of a uniform nature are just as simple as one another. Another kind of mind which might form an equally clear mental perception of some property of any one of these curves, as we do of the congruence of a straight line, might believe these curves to be the simplest of all, and from that property of these curves build up the elements of a very different geometry, referring all other curves to that one, just as we compare them to a straight line. Indeed, these minds, if they noticed and formed an extremely clear perception of some property of, say, the parabola, would not seek, as our geometers do, to rectify the parabola, they would endeavor, if one may coin the expression, to parabolify the straight line

    • James Franklin

      Interesting quote, but since then we've discovered vector geometry, surely the God's eye view. Straight is privileged in that geometry.

      • Commander Howdy

        Hi James, thanks for replying. Have you got a moment to explain: how is vector geometry a God's eye view?

        • James Franklin

          Maybe that's an exaggeration. But there's something remarkable: we take the continuum (a natural and numerical object: just all the possible ratios), take its cross product two or three times, and what we get is 2D or 3D geometry. Euclidean geometry,

    • Jimmy

      On another planet where vision never developed, and the intelligent sentient beings have senses that are as fine as electron microscopes, could geometry as we know it ever arise, and our maths be simply a quirk of carbon chemistry on this planet ?
      Or, is the physical world made in such a way that all intelligent life must necessarily develop with similar sensory properties, and come to perceive in some way lines, angles, curves etc. ?

      • Commander Howdy

        I think the point that Boscovich was making was that the underlying structures of the universe that Maths manifests could in another intelligence be otherwise described. That sounds, on the face of it, a rather bland thing to say, but on closer inspection I think Boscovich's remark was intended as a warning against intellectual hubris. Just because we see it as so, don't make it so.

        The second part of his great work has often been dismissed by contemporary scholars as a sop to religious oversight of his work: I sense that this opinion is wrong and that this enormous talent was to the end, a brilliantly humble man.

        • Jimmy

          It does sound interesting.
          I don't think lines as he describes them exist, though, I don't think anything in nature is uniform, or straight - except to our sense which, OK, is what he is saying.
          Do you think our perception of the straight line is bound to be abandoned in favour of the fractal ?
          I looked in to this a little when I did complex systems MOOC.
          Considering the simple pencil "line" drawn on paper which we are all familiar with, I did some googling, and found that paper has a fractal structure at the microscopic level, I think graphite does too. So enhance the senses with some magnification and the line disaapears and becomes something rather more complex - until it all disappears at the atomic level - and even then you may be left with fractal subatomic structures (?)(I didn't check this bit).
          So, perhaps it not such a bad notion to divide the ideal line into infinitesimals after all - in a way - the ancients may have been thinking along the right lines - as it were.
          Seeing as the brain also has a fractal structure, are the conceptions we have also of a fractal nature, and is this why the idea of a line being infinitesimally divisible was intuitively attractive to the first geometers ?
          I'm out of my depth here.

  • Leonie Green

    Succinct and challenging for the lay reader.

  • Roy Niles

    To James Franklin: That's a rather pointless answer below, since without the strategies that tell you how and why to build the bridges, including how humans have been devised to walk, crossing any twice would never need to be considered.

  • BDewnorkin

    Franklin claims that nominalism is wrong because it's intuitively evident that "pure mathematics reveals to us the topography of a region whose truths pre-existed our investigations and even our language" and it "reduc[es] mathematics to trivialities." The latter point is no argument at all and can be dismissed. The former point is useful for considering Franklin's article, more broadly. Intuition of the kind described is a popular but unconvincing premise for a metaphysical argument. It may well be intuitive for someone who acknowledges human intellect's capacity for "proof" and "visualization" to think that all mathematical properties are but formulations of this intellect, à la nominalism. This leaves us with nowhere to go.

    Similarly, that mathematics developed from symbolic formalizations of observed phenomena may prompt an inclination to include in the metaphysical catalogue of reality a space for mathematical properties. If left to stand alone, this intuition is feeble.

    Editorially restrained, Franklin's article would benefit from a more rigorous engagement with recent developments in metaphysics (which he no doubt does in his book). Bereft, it remains little more than a survey of three major positions in the philosophy of mathematics.

    • James Franklin

      Scare quotes around "proof" are not a good start. Proof is sacred (by the standards of mathematics) - and that doesn't mean derivation from arbitrary axioms either, it means demonstration from basic truths, a la Aristotle. I hope the book does engage with relevant recent metaphysics; obviously that can't be done in 2000 words.

      • BDewnorkin

        In both instances, I'm using quotation marks not as scare quotes but for quotation.

  • Roy Niles

    The contemporary scientific view has presented us with mathematically precise “laws” of the allegedly dead universe that nevertheless produce forces that appear to accidentally operate on a random basis to construct intelligently operated machines that consistently make successful independent “living” choices.
    Few seem to recognize that the mathematically precise measurements which have been proposed as descriptive of all universal elements cannot have been derived by accident, nor have acted to construct the functional systems of that same universe by applying some form of accidentally directed skills.

    Were they, then, derived from some version of a god? That makes even less sense, since gods supposedly made men to think uncertainly. Did these universal systems then evolve somehow? Most likely, yes. From what, we have no way of ever knowing to a mathematical certainty.

  • selimibn

    I disagree with the viewpoint expressed in this article. It would seem to imply, for example, that the geometry of manifolds in more that four dimensions has a different epistemological status than euclidean or Minkowski geometry, even though the arguments used to derive results in one and another are completely analogous. Or that complex analysis is somehow less real than real analysis -since, after all, no physical apparatus measures a complex quantity. Eggs stacking uses integer numbers, and yet we can reason and prove results about octonions. But one could certainly conceive of a world where eggs stacked like octonions. Physics tells us which one is the case; mathematics, by itself, cannot. If anybody thinks otherwise, I would welcome his opinion on the fundamental gauge group of particle physics. Can one, out of mere mathematical necessity, conclude whether SO(10), SU(5), or E8 is realized at high energies? Mathematics allows the description of the world that is, but also of countless worlds which aren't; and it doesn't contain a guide for picking the one we live in.

    • James Franklin

      The viewpoint of the article doesn't imply those things. That's like saying "It's a contingent matter whether my shirt is orange, therefore it can't be necessary that orange is between red and yellow."

      I don't think there are worlds where you can count with octonions, or where (possible) symmetry isn't described by the same groups as in this world.

      • selimibn

        I don't think there are, either, but I also don't think that tells us anything about octonions. If we adopt a philosophy of mathematics where we discover truths about the “mathematical properties realised in the physical world”, would it not treat differently the mathematical properties not realised in the physical world? Are mathematical truths about Euclidean space different from mathematical truths about fourteen dimensional non orientable manifolds?

        We live in a world that exhibits irreversible processes. Does that mean that the mathematical structure realised in the world is not a group but a semigroup? Would that change the formalization of our intutions about symmetry in the structure of a group or our understanding of the Poincaré group?

        • James Franklin

          Studying the mathematical properties realised in the real world is not the same as studying which mathematical properties are in fact realised in the real world. Compare: the science of colours studies colours, not whether some shade of blue happens to be uninstantiated. Mathematics studies all the mathematical structures. It's for the contingencies of nature (or the will of God, or whatever decides these things) to decide which are instantiated.

          • selimibn

            I agree that studying the mathematical properties realised in the real world, while also studying the mathematical properties not realised in the real world, and without making a distinction between the two is the same as studying mathematical properties, full stop. That indeed was my point, and that is what I always thought mathematicians did. However, I find it difficult to reconcile that position with the idea that some mathematical truths are known by perception. This seem to draw a distinction between instantiated and uninstantiated mathematical properties that I don't think is useful or tenable for mathematics.

          • James Franklin

            Some mathematical properties are instantiated, some not. Perception can obviously only access the instantiated ones. As described in the article, when looking at a regular 2x3 grid, you can see that 2x3 must equal 3x2. Perception is smart.

          • selimibn

            Yes, but if perception via instantiation has no bearing on the truth of mathematical statements, what's the point? And if it does, wouldn't you have to say that statements about instantiated properties and about uninstantiated properties are a different kind of true (or false)?

          • DanLanglois

            Hi selimibn, I think you and I are pursuing a similar point, here. I think we'll agree, that mathematics does deal with properties and relationships, patterns and structures. Now, here I want to go very slowly. What if I try to hypothetically add to this, that these are things which can be *instantiated* by many, very diverse things. This gets us into this notion that we 'decide which are instantiated'. Let me add another piece of jargon, here. The *universals*, discussed in mathematics, are *instantiated* by physical things. --physical objects. And also, maybe by physical properties. Like, lengths. Areas, volumes, velocities, accelerations, forces, masses.

            So we have *physical quantities*, and they are *instantiated* by physical objects, and these quantities, are in turn, what? Well, they stand in relationships of proportion to one another. Relationships of proportion hold, shall we say, between physical properties. Then, I might add here, that such relationships are 'physical'. I mean, perhaps, in some sense -- some legitimate sense. Not that I'm arguing for this. Maybe these relationships have a very important place within physical theory.

            So, I've introduced the term 'universals'. And these, need not be located in just one place at a time. They can be no one place in particular. I'm not sure we can all agree as to whether they can be nowhere at all. But, they can be in several places at once. Personally, I might ask what business sets, for example, have, being construed as physical properties/relations at all. Shall we say, that we are having our attention drawn to ever more 'abstract' properties/relations/patterns/structures.

            We're getting into properties of properties, relations among relations. Call this an upward march, up a hierarchy. Coextensiveness is the source of sets. Sets boil down to aggregates of individual essences. But, sets are not simply something that two coextensive universals share in common. Coextensiveness is a relation between properties of their members. What coextensive universals share, is a plural version of the individual essences of the instances. Anyways, I am hoping to make a hash of empiricist scruples, with this discussion, but I'm too lazy to keep dialing out if I haven't already won that argument..

            It seems to me that though the article rejects 'Platonism', it actually embraces what I might term 'epistemological Platonism'. Then, the difficulty (why it can have little chance of being true) becomes, as you have pointed out, the point that mathematical entities exist which are distinct from 'those of the physical world'. Somehow, you and I are wrong, I take it, in doubting that 'perceptions', of any kind, could result in the observation of objects that actually play no causal role, as it were, within the natural world of space and time.

            That is, we have a statement. And, it has its names, its quantifiers, its predicates. However, we are certainly not dealing with rude physical objects. Like, bodies, concrete individuals. And, the physical relations obtaining amongst them. However, we need whole systems of abstracta, for geometry, topology, number theory. So, maybe, we are supposedly abstracting on abstracta. And, we can comprehend the resulting abstracta, as a more radical one, starting from the original concreta. I'm thinking of maybe how to arrive, here, at anything like graphs, for example, by abstraction. Can we abstract from relations, other than from individuals?

            There are problems that I haven't even mentioned, that stand in the way of allowing abstraction to provide us with mathematical objects. What are large infinite cardinals? I think, upon some reflection, it ought at least to be clear that adopting the attitude that we are concerned with *actual* structures, here, is one of two things: hero, or ostrich.

          • James Franklin

            That discussion about universals and properties of properties is all good; maybe a bit confusing as to how it relates to sets, since sets are particulars, not repeatables.

            Some story like that about abstraction is no doubt right; unfortunately, there's no good account of the mental operation of abstraction.

            Everyone needs a story about how knowledge of large uninstantiated structures such as infinite cardinals can arise from our ordinary knowledge of finite numbers.

          • DanLanglois

            By repeatables I guess you mean "universals" or "types" that that particulars or "tokens" share. In which case, okay, blue is a universal but the set of all blue things is a particular? But, sets and relations are part of the background on which modern mathematics is built, and as such takes part in essentially everything you do. Boolean Algebra has applications all over the place in digital electronics (almost surely your personal computer got built on this basis). One might also find it noteworthy to mention that (classical) set theory has a deep connection with Boolean Algebra in that the algebraic properties of (classical) sets, and the identities of Boolean Algebra match exactly. So, one might point out that something very related to set theory has heavily practical applications. A little more concretely, one might point out the simplification of circuits which happens in Boolean Algebra, but explain that since you aren't going to talk about Boolean Algebra, you'll do such a simplification using the algebra of sets. In this way you can make simpler circuits which do the same things as more complicated circuits do by applying set theory.

          • DanLanglois

            'there's no good account of the mental operation of abstraction'

            It is 'abstraction' that is not a good account of the mental operation. Abstraction is logic, of which there is a perfectly good mental account.

          • James Franklin

            What is "bearing on"? Ontology or epistemology? My perception of the table doesn't "bear on" the existence of the table, it just tells me of its existence. Orange's being between red and yellow is just one kind of fact, whether those colours are instantiated or not.

          • selimibn

            Well, in my opinion it has bearing in neither. A previously unknown phenomena involving physical objects is not going to suddenly make a theorem about abstractions false. Rather, it is likely to change our mind about whether some particular aspect of reality embodies some particular mathematical properties. Maybe in some approximation a table is like R². But I don't think I can prove theorems about R² by looking at tables, and when I find tables are not featureless and have borders I will say that tables are not really like R², not that R² is not like we thought it was. In your yellow-orange-red example, do I learn something about the continuity of the real numbers by finding a wavelength between any two wavelengths? What if there is a fundamental width to color and suddenly I don't? Does that say anything about real numbers? Or will I conclude that the color spectrum doesn't really instantiate the real numbers? And again, if some mathematical truth is grounded in perception, how do I avoid the conclusion that perceptible and imperceptible mathematical truths have a different epistemological status?

  • DanLanglois

    What I find most striking among the comments, is this: 'I don't understand why you think mathematics cannot be done by "dreaming up axioms and deriving theorems from them".' Which received this reply: 'Can you name then any example of playing with arbitrary axioms that then turned into accepted good mathematics?' To which I myself might reply by simply saying 'all accepted good mathematics'. Then, I suppose that I have failed to distinguish notions like 'line' and 'point' from arbitrary axioms. But, indeed, they are worse than artitrary, they are notoriously difficult to define, which gets us into the (utterly necessary in all mathematics) concept of 'primitive notions'. --undefined concepts. Only motivated informally. Now, one might quibble w/this, and say that they are at least usually motivated by an appeal to intuition and everyday experience. I can, however, distinguish this from your 'Aristotelian realism'.

    'There are instances where topics of initially only pure mathematical interest turned out later to have applications (e.g. factorization with large primes in crytography), but that's quite different from playing with arbitrary axioms.'

    What about non-Euclidean geometry, used in Einstein's General Theory of Relativity? At the very least, I'm wondering if I understand your notion of 'arbitrary axioms'. Here, we're talking about setting the parallel postulate aside. We're saying that we'll consider two straight lines, indefinitely extended in a two-dimensional plane, that are both perpendicular to a third line. And, what is arbitrary here, is do they "curvee away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular? Or, maybe, they "curve toward" each other and intersect. Or, shall they remain at a constant distance from each other even if extended to infinity. These last, are known as parallels, so that one of the three options that I have given, is Euclidean geometry. I've described, here, merely two varieties, btw, of nonEuclidean geometry. I'm thinking, also, of affine planes associated with the planar geometries which give rise to kinematic geometries, if I may just recall this stuff to mind..

    • James Franklin

      Setting aside the parallel postulate was not arbitrary. It was motivated by the lack of total self-evidence of Euclid's fifth postulate, which referred to indefinitely distant space. So people wondered if you could replace that axiom with something else. It was a surprise that it turned out to have empirical application, but still, arbitrary it wasn't.

      Affine, projective etc geometries aren't arbitrary either. They're motivated by trying to find a skeleton or core to geometry: projective properties are more basic, somehow, than affine or metric ones.

      • DanLanglois

        I don't call it a surprise when math has an empirical application, and I've gone ahead and dialed out at length, above, about my take on whether setting aside the parallel postulate was 'not arbitrary'. To illustrate further, suppose that we're looking at Euclid's axioms. And, in a fit of attempted arbitrariness, I'm going to just add an axiom: A line entering a triangle at a vertex must intersect the opposite side, and a line that intersects one side of a triangle at a point other other a vertex also intersects a second side. In other words, if a line intersects a side of a triangle and does not intersect any of the vertices, it also intersects another side of the triangle. --not that I actually dreamed this up. Actually, some opinions (about deductive/axiomatic mathematics) typically attributed to David Hilbert can be traced back to the guy who did: Moritz Pasch. Let me split the axiom, into two axioms: In any right triangle the hypotenuse is greater than the leg. And, if triangle AOB is right, B lies between O and C, and D is the footpoint of the perpendicular from B to AC, then the segment OA is greater than the segment BD.

        So what? Well, these matters may have been considered too obvious, but the result of such neglect is the need to refer constantly to intuition, so that the logical status of what is being done cannot become clear. Doing "geometry without Pasch's axiom" begs the question of what the *other* axioms of geometry are. --Let me bring this down to Earth: the product of two positives is positive is essentially Pasch's axiom. Of course, the product of two positive real number is positive (positive times positive is positive), but when doing axiomatic geometry you don't know that you're dealing with real numbers. Let me give something else that is equivalent: A set S is convex if for every two points P and Q in S, the segment PQ is a subset of S. Now, okay, here is something like your ingenuous idea: Lines, planes, and so forth are idealizations of objects known from experience of the physical world. For example, a line is an idealization of a piece of string streched taughtly between two points. Similarly a plane is supposed to be a flat surface, like a table-top, but also is supposed to extend indefinitely in all directions. Sort of like, Nebraska. But, larger..

        ...we don’t have any direct physical experience with flat surfaces of indefinite extent. What I might be getting into here, is I might draw some pictures of convex and non-convex subsets of a plane. I shall do this, while keeping in mind that these are undefined terms:

        point, line, incidence, betweenness, congruence

        And I'm wondering, for example, can I guarantee that if a line enters the interior of a triangle through one of its vertices, it must also exit through the opposite side? Or heck, wait, what about if a line simply enters the interior of the triangle, then must it also leave it? The questions multiply, actually. If a line does not intersect either of two sides of a triangle, then does the line also not intersect the third side of the triangle? If a line intersects a triangle, then does the line intersect two sides of the triangle? Suppose that the line intersects the interior of a side of a triangle..then does the line intersect another side of the triangle?

        Now, I want to get into how it was once, and still is, worth debating a question like whether an axiom asserting a statement contrary to 'the true nature of space' would seriously compromise geometry's accuracy as a physical description. This is your idea, that geometry is a study of forms occurring in the physical universe. Aristotle's idea, if he has no objections. Forms occurring in the physical universe, or abstractions thereof. Theorems deduced from basic self-evident truths. You offered something about 'the lack of total self-evidence of Euclid's fifth postulate, which referred to indefinitely distant space'. However, what happens, when one has become accustomed to distorted representations, and learned to read them comfortably? Nevertheless, the one sacred relation that you shall always depict accurately, is lines that do, or do not, intersect one another will faithfully appear that way on the page.

        My point, is not "let's give harmless nonEuclidean toys to mathematics majors'. I'm talking about something of great psychological importance that you can afford to lose.

  • DanLanglois

    Also, in the penultimate paragraph you own this question: 'What is necessary is true in all possible worlds, but how can perception see into other possible worlds?' However, previously you had said of logicism, that it 'has not been accepted by any serious philosopher of mathematics for 100 years.' You quote Peter Singer for a description of this philosophy of mathematics: ‘The self-evidence of the basic truths of mathematics,’ he says, ‘could be explained… by seeing mathematics as a system of tautologies… true by virtue of the meanings of the terms used.’

    Here, my point is not to disagree about logicism, though I think you may overstate your point. 100 years ago it was 1914. Nevertheless, my point is that you've rejected the idea of seeing mathematics as a system of tautologies, and I think that entails rejecting the idea that 'What is necessary is true in all possible worlds'. You'd be right about what is *logically necessary*, but not about the certainties of mathematics, which may quite coherently be regarded as not true in all possible worlds.

    • James Franklin

      I don't really follow that. If something is necessary, surely it's true in all possible worlds? Many philosophers regard that as the definition of necessary (e.g. Lewis's possible worlds theory).

      Do you have an example of a "necessary" truth of mathematics and a possible world in which is isn't true?

      Re history: by 1910 even Russell had some to accept that the Axiom of Infinity (that the numbers don't run out) wasn't a matter of logic (though an axiom essential to mathematics).

      • DanLanglois

        'Do you have an example of a "necessary" truth of mathematics and a possible world in which is isn't true?'

        Do I have an example. Yes, all geometrical theorems are derived from at least one 'not logically true' axiom. Perhaps I am not being generous to allow that this is not common knowledge, but I hope that you're keeping an open mind, at least. Let me put a pin in that, and try a simple example first..

        Let me translate "15 + 2 = 17" into predicate logic notation. My idea, is to state an arithmetical truth in the notation of modern predicate logic, and then test its logical truth. The left side of the equation, "15 + 2", contains a function. Let me restate it thus: *add* (15, 2), using standard function notation (which is part of predicate logic). Since *add*(15,2) resolves into one number (or object), the expression behaves like an individual constant. It is the sort of thing which may be used as the argument to a first-order predicate. And, equality is a two-place predicate.


        Let me just note here, that "7", "5", and "12" are symbols —numerals, not numbers. My idea, is that they are individual constants which stand for objects from my domain, which is the domain of natural numbers. That way, I cannot know whether *Equal*(*add*(15,2),17) is true or false until I assign values to these currently meaningless symbols. That this:


        And, I know that if I assign the sequence of values, , to that sequence of symbols, then the formula comes out true. The formula is also satisfied by and by . So it is satisfied out of my domain more than once. But it is not satisfied by or by . Hence the formula is not satisfied by all sequences in the domain, and hence is not true for this interpretation. Since the formula is false for at least one interpretation, it cannot be logically valid.

        Satisfaction is the weakest level of truth. Truth for an interpretation is stronger, and logical validity (or logical truth) is the strongest. If at least one sequence of objects in the domain, when assigned to the sequence of individual constants in a formula, makes the formula come out true, then you say that the formula is satisfied. That is, a formula is satisfied or "true for some sequence" when it is true for at least one sequence of objects from the domain of an interpretation. If the formula is true for every interpretation, then it is logically valid.

        Then, arithmetic expressions are satisfiable at best, and not logically valid. And, this is shown e.g. in the ways in which logical systems must be extended to permit the derivation of arithmetical truths. We may still be reeling at this 'surprising conclusion', but consider, that a formal system of predicate logic with nothing but logically valid axioms can be consistent and complete in the sense that all logically valid formulas of predicate logic will be theorems of the system. Then, all the theorems of such a system will be logically valid. And, this is a great strength. However, it also proves that it is not yet adequate to capture arithmetic, since arithmetical truths are not logically valid. To extend it so that it does capture arithmetic, we must add axioms which are not logically valid. Only then will we be able to derive theorems which are not themselves logically valid.

        What then, about all geometrical theorems being derived from at least one 'not logically true' axiom? I have a chance, here, to follow up on my comments about how mathematicians were perturbed by the parallel postulate. It seemed true. But, perhaps two consistent geometries could be developed? A non-Euclidean geometry, containing its negation as an axiom? And this gets us into the success (demonstrated consistency?) of non-Euclidean geometry.

        Both the affirmation and the negation of the parallel postulate led to consistent geometries. But this means that the parallel postulate itself is not logically valid. This is because its negation does not lead to contradiction. If it were logically valid, then its negation would be a contradiction and could not be an axiom in any consistent system. But, it could be an axiom in a consistent system. Hence, the parallel postulate is not logically valid. Also, each of the original Euclidean axioms, not just the parallel postulate, is independent.

        Also, non-Euclidean geometry is actually, only the first case in the history of mathematics in which an axiom of a consistent theory was replaced by its negation without introducing inconsistency. That is, it went down like this: perhaps a consistent non-standard arithmetic could be created by negating a standard arithmetic axiom? And, this turned out to be possible, and before long non-standard theories were the subject of serious study in arithmetic, geometry, logic, set theory, analysis.

        And, underlying them all is a principle that if a given proposition is not a theorem of a system, then its negation can be added to the system as an axiom without creating inconsistency. And also, once you know that a proposition is 'undecidable', --when neither it nor its negation can be deduced from the axioms of that system, then you know that that either it or its negation can be added to the set of axioms for that system without introducing inconsistency. And, meanwhile, every sufficiently powerful, consistent axiom system of arithmetic will contain undecidable propositions. In particular, some of these undecidable propositions will be truths of arithmetic. This is under Gödel's theorem, of course I mean first incompleteness theorem. There is maybe no such thing as a quick overview of Gödel's proof. I think I'll just assert that the confirmation of my point is quickly derived. Arithmetic truths not logically true.

  • Maggy Franklin

    As an educated "lay-person" I thank you very much for the examples of the ideas expressed - they make what you say very readable. I'm well aware the summary is only merely scratching the surface but it gives one a glimmer of the depth of the concepts. To anyone who hadn't thought about "mathematics" this is would be a wonderful introduction to questions they didn't know existed!

    • James Franklin

      Very good. There's plenty out there to think about ...

  • Jimmy

    Slightly off topic, hoping that people are still reading, but I'd like to ask a question or two.
    If evolution is the overarching mechanism by which biological life is steered, and mathematics is the product of an evolutionarily developed brain, why should we expect any sort of truth from mathematics ?
    That is, shouldn't mathematics just be seen as another manner in which organisms seek to flourish and reproduce - and that requires no great truth of the universe to be revealed ?
    I think there is an evolutionary psychology view of religion in which it is held to be valuable as a social mechanism, helping the continuance of the species - even while most religious premises are held by science to be patently false.
    There is an assumption, isn't there, that science is part of an intellectual evolution that has moved us on from ignorance to truth, but do we not thereby introduce a sense of revelation where it doesn't belong ?
    And if mathematics is merely a phenomena of utility to breeding and survival, not a phenomena of revelation of truth and insight, should we not expect it to eventually disappear or become something else if it loses that utility ?

    • James Franklin

      You might as well say, "Perception is the product of an evolutionarily developed brain, so why should we expect any sort of truth from perception?" There must be something wrong with that. It's an example of the argument that David Stove identified as the winner of his competition to find the worst argument in the world: the argument: We can only know things through our forms of perception/evolutionarily developed brain/education/culture, therefore we cannot know things as they are in themselves.

      • Jimmy

        You ask that too. I thought that was accepted as the way things are. Mathematicians and scientists are not exempt from the process of evolution are they - especially those doing it professionally ? Why would evolution grant truth to Darwin and Russell, after billions of years of just giving stripes to zebras and sonar to bats ?
        OK I skimmed through the paper, and you agree that there are a whole bunch of subjectivites that hold us away from seeing things as they are, to one degree or other, but that 2+2=4 whatever the subjectivities.
        Will 2+2=4 still be here in a million years time, or 100 million ?

        What if I were to be an evil scientist, employed by Orwell's Miinistry of Truth, and locate the exact neurons that tell us 2+2=4, and the genes needed to develop those neurons, and make a virus which alters those genes at the germline level thus breeding out the ability to know 2+2=4, and rewire them to tell us 2+2=5. Finally I destroy those neurons in my own brain to remove the last vestiges of 2+2=4.
        Would it then still be true ? Is 2+2=4 really so fundamental that it doesn't depend on brain tissue at all ? Or so fundamental that it cannot be hacked in such a way ?

        • James Franklin

          The Ministry of Truth can make everyone say that black is white, and maybe make everyone believe black is white, but it can't make black white.

          • Jimmy

            I would think colour would be even easier.
            I'm not convinced about this, I'm sure I read that there are neurological structures (I just checked, there are) that are involved in defining the sorts of boundaries between objects - that are essential (seems to me) for having the basics of number. So let's have a virus that just messes those brain cells up.
            Why should I assume that number isn't an arbitrary phenomena, based on the limited working of neurons applied to a real world that isn't really divided in the way our brain says it is ? A little like the way we have a limited range of colours to perceive ?

          • Jimmy

            I would think colour would be even easier.
            I'm not convinced about this, I'm sure I read that there are neurological structures (I just checked, there are) that are involved in defining the sorts of boundaries between objects - that are essential (seems to me) for having the basics of number. So let's have a virus that just messes those brain cells up.
            Why should I assume that number isn't an arbitrary phenomena, based on the limited working of neurons applied to a real world that isn't really divided in the way our brain says it is ? A little like the way we have a limited range of colours to perceive ?

      • Jimmy

        James, did you ever see any of this ? I just found it and it goes into some of the things I was thinking about - though have yet to find an answer to the question of whether the basic objects of geometry are some of evolution's useful little fictions.

        • James Franklin

          Looks like idealism dressed up in fancy equations to me.
          The desktop analogy is quite interesting; but there's something wrong with it. If the file icons on the desktop are blue, it doesn't mean the files are blue, but if the file icons are 10, the files are 10 (or should be). It's something like perception of primary vs secondary qualities: things literally have shape, size and motion something as we perceive them, even if they don't have redness as we perceive it.

      • Jimmy add, Hoffman discusses maths here -

        - and concurs with yourself I think that 2+2=4 , always.
        However his whole work starts off using maths, so I would expect it to remain valid throughout.
        Hmmm.... :-)

  • Vadim Kataev

    In mathematics, like in a perceivable world on our physical scale, we have stable objects. It is easy to see how for example the addition operation is related to our spatial intuition, to perceivable truth about "whole as the sum of its parts", to the principle of locality. It is also not difficult to imagine another worlds, where the principle of locality is not valid or where no conservation laws exist.

    • James Franklin

      Failure of conservation laws doesn't imply failure of mathematical truths. For example, if I put together two raindrops and end up with one because they coalesce, that doesn't mean that in the raindrop world 1+1=1. "1+1" refers to distinct objects.

  • skanik

    I think the explanation of Platonism is a bit off the mark.

    When you gather five sticks together - how do you know there are five sticks
    and that those are the only sticks you should gather into a pile for counting ?

    You make an inference.

    Something the mind does naturally and cannot be programmed into a computer
    or explained in any other way.

    In Platonism you recognise a pattern of Truth because the Truth is inside your mind.

    • James Franklin

      Platonism is a theory about ontology, not epistemology. It says there are Numbers etc. So how can they get inside the mind? And how can they have relevance to your counting sticks?

  • PeacePoll

    I'm late to this discussion - I hope someone else is hanging around!

    The Nominalist view seems unlikely to me. Mathematicians take simple starting points and construct concepts that have no connection with reality – until one day, it is discovered that they do. Eugene Wegner wrote of this in his paper "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". Mathematics was not created to explain the world around us; instead, that world is found to match mathematics. This is a very peculiar state of affairs.

    But it is not just this uncanny matching that is strange – it is the fact that there is any matching at all. Why should gravity obey an inverse square law? Why is road damage proportional to the fourth power of the axle weight? The physical universe and mathematics are inextricably intertwined, and I suggest that the two are in fact one – that the physical universe is primarily the manifestation of mathematics.

    But it is not just this uncanny matching that is strange – it is the fact that there is any matching at all. Why should gravity obey an inverse square law? Why is road damage proportional to the fourth power of the axle weight? The physical universe and mathematics appear inextricably intertwined, and I suggest that the two are in fact one – that the physical universe is primarily mathematics, and our experience of the universe is how mathematics manifests itself to us.

    That is a large claim, so I shall start small. For an electron or a photon, there is no difference between the particle and its mathematical description – the particle is its wavefunction. Furthermore, we experience photons directly. Let me emphasize this: we are able to experience a mathematical description directly. But larger objects also have a wavefunction, even though at the macroscopic level, the equations are impossibly complex. Perhaps the mathematical attributes of macroscopic objects can be distilled from such complexity, rather like a Taylor series reduces to a simple function.

    One interesting wrinkle arises from this conclusion. Consider Gödel's Incompleteness Theorem, which states that in a mathematical system of sufficient complexity, there are propositions which cannot be proved true or false. (The system can be extended to cover these propositions, but further undecidable propositions will always arise.) So there must be aspects of the universe which cannot be treated mathematically.

    Now mathematics is the sine qua non of science. An event that cannot be counted, weighed, timed or otherwise measured is not amenable to the procedures of science; it is a one time event, a fluke, a chance confluence of nature. Science only considers phenomena that can be treated mathematically in some way. That is to say that science is necessarily an incomplete explanation of the world, and there will always be areas where it cannot tread.

    Disclaimer: I am neither a professional mathematician nor physicist. If you are, feel free to slap me down and point me in a useful direction.

    • James Franklin

      Well yes, more or less. It's true that most of what's observed in physics and other sciences is really the mathematical properties of things. A couple of clarifications could be: surely Wigner goes too far, since a lot of mathematics is really to describe mathematical properties, so it isn't surprising they apply: for example, although it's a mystery why gravity obeys an inverse square law, it's easy to see why light intensity as a function of distance from source obeys an inverse square law.
      I wouldn't like to reify a wavefunction. That would be like saying a physical object consists only of its length. The wavefunction describes the distribution in space-time of some quantity (hard to say exactly what it is).

      • PeacePoll

        Oops - Wigner. I must have had Daniel Wegner on my mind.

        I take it from your comment about reification that you don't agree, and I am not trying to enter into a long harangue over this, but I wanted to point out some aspects of such a claim.

        Yes, certain properties like light intensity follow from the geometry of things; it's the deeper characteristics that are puzzling. The conventional view is that the observable world is made of stuff which has mathematical properties inextricably bound up with it. The nature of this association is akin to the mind/body problem, which is commonly resolved by asserting that they are two aspects of the same thing.

        I suggest a similar resolution here, that the world /is/ mathematics, and we experience it as objects etc. due to our limited understanding. They are metaphors for the math, rather like the hydraulic metaphors used to explain electricity.

        Now I agree that this is all very fanciful and a long way from the common experience. My proposition is quite possibly undecidable, as I cannot think of any experiment that would satisfy the logical positivists, but it strike me as eminently logical, satisfies Occam's razor, and by invoking Godel, leaves room for the ineffable. (Here I would place consciousness and qualia, though that leads us off-topic.)

        • James Franklin

          Both of those suggestions (mind/body are aspects of the same thing; the world *is* mathematics) seem to me category mistakes, much like saying that the colour of an object is an aspect of its shape. Things in different categories aren't candidates for identification, so something in the category of substance, like the world, can't be identified with a property, like quantity.

          • PeacePoll

            Thanks. I'll ponder that.

  • YF

    In mathematics, 'truth' is relative to a particular set of axioms (which may or may not map onto nature). So even in math there is no such thing as 'necessary truth'. While I applaud the authors' defense of Aristotelian realism, his contention that this position somehow presents a challenge to naturalism is unfounded. Does the fact that we cannot yet create an artificially intelligent mathematician count in favor of supernaturalism? Indeed, how would supernaturalism provide an answer? Is it that unreasonable to imagine that mathematician brains are the product of evolution, development, and learning (education)? Furthermore, it's worth pointing out that the authors' position (Aristotelian realism) is not new but was articulated many years ago by physicist Simon Altmann in his book, Is Nature Supernatural? and by mathematician Morris Kline in his book, Mathematics and the Physical World.

    • James Franklin

      Mathematical truth is not relative to particular axioms. That's a philosophers' misconception of mathematics. "There are six pairs in four objects" is just true - check it out for yourself - and can't be changed by choosing other axioms.
      Of course mathematicians' brains are the product of evolution and learning. The problem is to explain how such a brain can have insight into necessary truths.

  • YF

    The 'necessity' of mathematics is only an illusion. Mathematics can be empirically true (in which case it is not necessary), or it can be formally true (in which case it is 'necessary' only insofar as it can be derived as theorems from a set of axioms). Please see Simon Altmann's chapter on this issue in his book, wherein he differentiates between 'empirical' mathematics and 'game' mathematics - both of which enter into the practice of mathematics, but are often not recognized as such or properly differentiated, leading to unnecessary angst regarding the (pseudo-problem) of the unreasonable effectiveness of mathematics in describing nature. His thesis is perfectly in line with your Aristotelian view, however there is no need to posit anything supernatural in accounting for the practice of mathematicians nor for the empirical applicability of mathematics in the sciences.

  • Dietrich Krueger

    Einstein wrote, "One reason why mathematics enjoys special esteem, above all other sciences, is that its laws are absolutely certain and indisputable, while those of other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts."

    The reason why mathematical laws are indisputable is simple: They do not talk about the real world. Nothing could ever happen in the real world that would make us refute a mathematically proven law. Math, like any logic, is not an empirical science. Math consists of axioms and tautologies, and that's why it is "true". To say that math existed before the human mind developed it is simply wrong. That we "discover" mathematical proofs for mathematical conjectures shows, in and of itself, nothing more than the good mathematical instinct of those who proposed the conjecture. Before somebody finds the proof, we simply do not know whether it exists. Statements such as, the truth already existed before it was proven, make no sense. The subject of math is quantity, a logical and not an empirical term.

  • James Franklin

    An Aristotelian Realist Philosophy of Mathematics is reviewed in the Sept 2014 New Criterion.

  • Patrick Lee Miller

    The argument of this article is premised on a false dichotomy: either mathematical properties have causal powers for perceptible things, and are thus in them; or they are not in perceptible things, and thus lack causal powers for them. It ignores the tertium quid: that mathematical properties have causal powers for perceptible things, but are not in them. After all, something need not be in something else in order to have causal powers for it. The sun is not in my skin, and yet the sun has causal powers for my skin -- for example, when it burns me.

    The article's critique of nominalism -- which presupposes the objectivity of mathematical truths -- sticks, but the critique of Platonism accordingly fails. This does not prove Platonism true, but it does undercut the argument presented by the article for the Aristotelian position, inasmuch as this argument relied on the rejection of the two alternatives proposed at the beginning (nominalism and Platonism). Deciding between Platonism and Aristotelianism requires another sort of debate, one that never arises in this article.

    This is the debate whether mathematical properties (or forms, more generally) are *in* perceptible things (also called particulars) or bear some other relation to them (typically, they are said by Platonists to be *above* them, because they care causally superior to them: that is, they cause them but are not caused by them). The argument against the Aristotelian position is quite straightforward. If forms were in particulars, they would not be repeatable in the way they are.

    Thus, if my penny is in my pocket, it cannot also be in yours; and if my penny is in your pocket, it cannot also be in mine. Correlatively, if a mathematical property (symmetry, twoness, or whatever) is in the eyes of my face, it cannot also be in yours (and vice versa). Yet these properties can be true of both our faces; they are repeatable (not just once, but in principle ad inifinitum). We say colloquially that they are both *in* our faces, but that is a metaphor. What it means, literally, is that those properties have causal power over our faces (e.g., *because* I have two eyes, I have an even number of eyes, and so on).

    The only philosophy of mathematics that survives these observations is Platonism: mathematical properties (and forms more generally) have causal powers over perceptible things, but they are not thereby in them (except metaphorically). Additional arguments -- sound ones, in my view -- conclude that in order to satisfy those criteria, these properties (forms) must be non-spatial and non-temporal, and thus inaccessible to our senses as such. This is not to say we do not perceive twoness in a way -- look around you, and you will see several perceptible things that appear to be two -- but rather to say that when we are being precise we must conclude that our eyes alone do not perceive twoness, they perceive perceptible things, which our acquaintance with twoness arranges into sets.

    This is all Plato, by the way, who seems to differ from the so-called Platonists in the philosophy of mathematics. For example, his forms are not "abstracta" (things abstracted from perceptible things, using an empiricist epistemology which draws al knowledge from the senses); it would indeed be odd to find such derivative things exercising causal power upon the perceptible things from which they were derived. On the contrary, Platonic forms are antecedent, as is our knowledge of them. Only so, he argues, can we account for the fact that we can do mathematics, or any other inquiry which delivers conclusions that are objectively true.

  • Jonathan S

    Hi James - very interested in your approach, I have been debating this question on forums. I too am a mathematical realist (even though I'm not much of a mathematician!) The way I have been putting it, is that 'number is real - for an intelligence capable of counting.' By this I mean that number is not internal to the process of thinking, that it is genuinely discoverable, not simply a projection or invention. The ability to perceive number (or ratio) is fundamental to the nature of rational intelligence (indeed is the very root of 'rationality'). I take this to mean that the rational mind can see a higher order or domain of reality, than the merely sensory. It is not separate to the sensory realm, but is what Aquinas would have called 'subsisent' within the sensory realm. Reason is the capacity to perceive such rational relations, which is one of the cardinal differences between humans and animals.

  • NicolasBourbaki

    "It has not been accepted by any serious philosopher of mathematics for 100 years."

    I know of two mainstream philosophers that has tried to revive logicism: Crispin Wright and Bob Hale.

    • James Franklin

      It's true that Wright and Hale call their position "neo-logicism", but they rely on a relation of pairing for equinumerosity of sets, which seems to me as non-logical as e.g. the Axiom of Infinity, whose non-logical nature was a main complaint against the original logicism.

  • James Franklin

    There's a related debate about infinity at

  • drgsrinivas

    Mathematics is simply a language which uses numbers and symbols to express.
    And it is basically built upon logic but not beyond logic.

  • James Franklin

    Aristotelians agree that mathematical truths are true independently of us, and that we discover, not create, mathematical truths. Platonists shouldn't maintain that those theses are unique to them. The difference is that Aristotelians point out that those truths, or some of them, are realized in the actual world.

  • James Franklin
  • Joseph Lammers

    I read Mario Livio's book, "Is God a Mathematician" about a year ago. Its central question was: is mathematics invented or discovered? My own view, which the book doesn't really answer, is that it is discovered.

    • Joseph Lammers

      I guess, in a sense, that would make me a Platonist.

      • James Franklin

        No, it doesn't make you a Platonist, it makes you a realist - meaning, you think mathematical truths are out there, not invented. Realism comes in two varieties, Platonism (meaning that the truths are in an abstract world divorced from the physical world), and Aristotelianism (meaning that the truths are realizable in the physical world).

        • Joseph Lammers

          Interesting take, and personally I think you're right. Livio seems to associate the discovered view of Mathematics with Platonism, however. It's been over a year since I've read it so I'd have to go back over to nail that down. Since I can't envision a math that postulated 1+1 = 3, for instance, would have any practical use at all I don't see how it can be considered to be invented. Perhaps in some of the higher levels of math, but then even branches of mathematics that were originally thought would never have a practical use turned out to be quite effective in various areas of science.

  • Albert H C Manders

    While I am not disagreeing with the "conclusion that pure mathematics reveals to us the topography of a region whose truths pre-existed our investigations and even our language", the example of the sum of digits divisible by 9, is a bad example as it is not supportive of this claim. The example works a result of 9 being one less than the base in which the numbers are expressed. It is in fact a logical triviality resulting from the symbolic language we typically use.

    • James Franklin

      Well, the same happens with any other base (for base n, it works with n-1 in place of 9, as you say). That's still a mathematical fact that's not obvious and needs proof. However, I agree that for the cognoscenti it would be better to take an example that's entirely free of the symbols we use for numerals. The logarithmic falling off of the density of primes is a good example of the mathematical topography that's out there to be discovered.

  • EmpiricalWarrior

    Well the runaway freight train of mathematical hubris is on full display here. Logical fallacies, baseless assertions and reified human concepts have all been loaded aboard and the whole groaning mess is rolling down the tracks of metaphysical self-certainty toward the boundless landscape of the human imagination and as far from reality as it can possibly get. Pop the warp drive boys and let's head on over to the next universe. I hear there's 72 virgins for every nerd!

  • Commander Howdy

    Faith, like I said. Peace.

  • James Franklin

    The ideas of the article are reviewed in Philosophia Mathematica.

  • Doc c

    We perceive the world mathematically because we evolved to perceive the mathematical world. Prawan Sinha's experiments with people who are blind from birth and then have their congenital cataracts removed shows that our abstract constructs do not cross from tactile to visual until those 2 senses are allowed to work together (they could not recognize shapes visually that they could by touch until they could touch and see at the same time). We cannot understand a universe, or parts of our universe that are not mathematical. Does that mean such entities do not exist? Aristotelian Realism reveals the limits of the human mind, and opens the naturalist door to a more hopeful imagination than that of pure reductive determinism.

    • James Franklin

      True. Multimodal perception is very interesting, as it must be of properties independent of any one sensory modality. (But "We cannot understand ... parts of our universe that are not mathematical" may be going too far: we understand, in a way, consciousness and felt forces prior to any mathematization of them.)

      • Doc c

        Perception, or behaviorally driven cognition is not the same as understanding. Our senses evolved in a mathematical universe, or part of a universe. Our senses constrain both concrete and abstract cognition. Only abstract cognition allows us to imagine a world without mathematics, but how can we even imagine the operations of such a world without sensory experiences of its operations to guide us. Touch cannot guide vision via any integrated abstract understanding. Vision must have the chance to experience what touch has experienced. Our integrated brain will not be able to understand something that any and all of our senses did not present to it. What non mathematical experience ever gets presented to our brains?

  • ZZMike

    PS: The old joke: The Universe is the way it is because the mathematics that describe it are true.