Mathematical aristotelianism

Aristotle's philosophy was a reaction to that of his predecessor and teacher Plato. One particular detail that separates the two philosophers is the question of the forms, or universals. While Plato claimed that the forms are eternal, existing independently of their physical actualisations, Aristotle thought that the forms were inherent in the objects, without any possibility of existing independently of the objects. Consequently, mathematical Platonism sees physical objects as instantiations of general forms, while mathematical Aristotelianism views mathematical forms as attributes of particular concrete objects. The question is, do particular objects (such as physical bodies) require the platonic forms for their existence, or do, as Aristotle claimed, the forms depend on particular objects for their existence?
At least when it comes to mathematics, the Aristotelian view runs into trouble because there are mathematical objects and results which cannot possibly have a physical basis. Let's list some examples:

• Infinite sets. Physics tells us that there are only a finite number of (observable) physical entities. (There may exist infinitely many distinct universes, but if you believe this, then you don't need to be convinced of mathematical Platonism.) Aristotle tried to solve this problem by rejecting the existence of 'actual' infinity, and replacing it by 'potential' infinity. It's not entirely clear whether this distinction makes any sense when analysed closer, but in any case modern mathematics could definitely not work without something like actual infinity. Moreover, it is a fact that modern mathematics has tremendous applications to all aspects of science. Hence mathematics is something that uses concepts without physical ground, and at the same time it is something whose consequences deeply affects physical reality.
• The Banach-Tarski paradox. If this result had a physical grounding, it would most likely have to be its standard interpretation involving two physical objects and decompositions of these. However, the mathematical result then implies something which is physically impossible, namely that one solid ball can be decomposed and the pieces rearranged so that to form two balls, each of the same volume as the first. The Banach-Tarski theorem can therefore not possibly have any physical grounding. Thus we have here another example of a mathematical form which cannot possibly be realised physically.

There are more examples of this kind, but it will not change anything to include these. To these examples the Aristotelian can reply that 'real' mathematics is grounded in physical objects, such as a number of pebbles on the ground, and then mathematicians dream up all sorts of abstractions on top of that, which are are just mental constructions with no connection to the real world. Well, the problem with this view is that these 'dreamt-up' abstractions have applications to the physical world which are just as intimate as the relation between three pebbles and the number three. To take a simple example, consider differential and integral calculus discovered by Newton and Leibniz. Operating with infinite sums of infinitely small quantities led to modern calculus and the continuum, techniques essentially incorporating infinite objects, and without which most of physics and its applications in technology, would not have seen the light of day.

Newton's platonism

The following often cited quote by Isaac Newton shows his thoroughly platonist attitude. Since he was primarily a theoretician it is safe to assume that the statement is as much about his mathematical discoveries as they might be about physical facts:

I do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.

It is interesting to note that another mathematical genius, Grothendieck, has also compared the mathematical body of truth, unknown to the mind, as an ocean.

Comments on Gardner and Davies

The most recent issue of the EMS Newsletter

http://www.ems-ph.org/journals/newsletter/pdf/2009-06-72.pdf

continues the series on Platonism with contributions by well-known Platonist and puzzle-composer M. Gardner, as well as another article by B. Davies whose first article initiated the debate and has been commented on in a previous post here.

Gardner's article is mainly a personal reply to Hersh and makes the case for 'Aristotelian Platonism' (yes, it sounds paradoxical), that is, the view that mathematical reality subsists in physical reality as forms of material objects. In this view mathematical reality is independent of human minds to the same extent that physical objects are so. I have already pointed out the inadequacy of this view in an earlier post, but will repeat some arguments in relation to Gardner article, because it offers a particularly good illustration.
Gardner says early on:

Consider pebbles. On the assumption that every pebble is a model of the number 1, obviously all the theorems of arithmetic can be proved by manipulating pebbles. You even in principle can prove that any integer, no matter how large, is either a prime or composite.

Already here there is something that is not right. Arithmetic reality cannot subside in physical pebbles because there are only a finite number of pebbles in the universe. If there were infinitely many pebbles, the universe would have infinite mass, and so would collapse into a singularity by the infinite gravitation. Therefore one can never prove anything about all (or infinitely many) integers and claim that this fact is a fact concerning physical entities. One can certainly not prove Euclid's famous result that there is no largest prime number by any manipulation of pebbles (or any other physical objects one happens to fancy playing with).
What Gardner's mathematical Aristotelianism leads to is mathematical finitism, that is, the belief that there are no actual mathematical infinities. Aristotle, and many finitists admit potential infinities, that is, patterns that in principle go on indefinitely, but which, it is claimed, do not actually go on infinitely. Personally I don't think the concept of potential infinity is well-defined, and it is not clear if it has any meaning beyond the muddy word-twisting of philosophy. How can we say that something in principle goes on forever, without it actually doing it? What gives us the right to assert this, and what does 'in principle' even mean here?
On the other hand, infinite sets have a precise mathematical definition which was given by Cantor, namely that to be infinite is to have a proper subset of the same cardinality as oneself.
In any case, modern mathematics could not exist in its present form without actual infinities. It is fair to say that actual infinity is indispensable to mathematics in the same way that mathematics is indispensable in natural science. To doubt the existence of something which is the basis for things one does in every-day life is a very dishonest attitude. This is an analogue of the Quine-Putnam indispensability argument for the reality of mathematical objects.

As I have written before, the Aristotelian view of mathematics is enough if one is only considering mathematics at the level of pebble counting or simple geometry. A similar mistake is made by conceptualists like Nunez who thinks that by explaining the axioms of mathematics in terms of cognitive structures, one has thus reduced all of mathematics to a mental construction. Gardner has no problem fitting the primality of 17 or Klein bottles into this picture, but already at the level of complex numbers and derivatives he begins to struggle, and has to resort to saying that these entities are probably somehow embedded into the physical universe, even though they do not have concrete material models. Already here we can notice a drift away from Aristotle who said that mathematical forms have to be carried by physical objects, to something more Platonistic, namely that the forms are somewhat more autonomous and that forms are instantiated by physical objects.
One can go even further. Try to explain how algebraic schemes or infinite dimensional representations of Lie groups depend upon the physical structure of the universe (and not the other way around), and how their existence depends on the existence of physical objects (or human minds!). Then explain how these abstract entities can play important roles in mathematical physics, which is a science describing physical objects. It is pointless to try to see these abstract entities as something residing in for example elementary particles. However, we know that certain abstract mathematical entities have a direct relation with these very same elementary particles, because we can describe the latter using the former.
In many ways I agree with Gardner's views, and it is commendable to try to defend mathematical realism in public philosophical debates using the common sense principle that existence is first and foremost material existence (a principle I don't agree with). Some people are using a similar approach in an effort to reconcile science with religion. The problem is that it doesn't quite work all the way. I have tried to explain above why it doesn't work for mathematics. Perhaps I will some day write something about why I do not think it can work adequately in the science-religion bridge building.

Regarding Davies' article, it mainly adds some details to his first one, and gives replies to the other contributions that appeared in this debate, noticeably the Platonistic one by Mumford. Davies mentions the non-Platonism of P. Cohen. This is a well-known theme basically dealing with a very special form of Platonism, namely set-theoretical Platonism. Platonists do not necessarily believe in the existence of sets, but a more reasonable view is the one mentioned in Mumford's article mentioned above, namely that set theory offers one possible model for mathematics. Mathematical facts can be grasped either through Russell-Whitehead's Principia Mathematica, or equivalently, though Quine's New Foundations. These are just two different formalisations of the same mathematical objects. If one is a formalist like Cohen professed to be in Davies' quotation, then one would regard the question of different formalisations describing the same objects as absurd and nonsensical. Still it is obviously the case that different models exist for the same mathematical entity. Set-theoretical Platonism may be wrong, but it is not the only type of mathematical Platonism.

Davies admires P. Davis' history of negative numbers and contrasts this with the discovery of the moons of Jupiter or America. The argument is that if it took such a long time for mathematicians to accept negative numbers, then it must mean that these are a social construction rather than a discovery of some objective existence. The point is however that there are many instances in the history of science where facts and discoveries have been accepted only after a prolonged debate. It took a while and heated debates before the existence of the vacuum was accepted in physics, and the debate continues with the latest findings of quantum physics. Today we are debating whether dark matter exists, or whether the anomalies in galaxy dynamics are simply due to inaccuracies in our physical formulas. How long did it not take until Darwin's theory was accepted. Is it even accepted now? Deep and complicated facts take time to digest and understand. Mathematical objects like negative numbers are abstract, so there is no surprise that it takes a while for them to be generally accepted. Of course mathematicians do not have a 'direct perception' of the platonic realm. We have to constantly push ourselves to our limits in order to get mathematical understanding and insights. Historically, only people who pushed their knowledge, perception and abstraction sufficiently, could grasp the negative numbers, but once the negative number were put into a mathematical and pedagogical framework it became much easier for subsequent generations to grasp them.

To compare abstract entities with very concrete ones like continents is like comparing apples and pears (even worse!). A future P. Davis appearing in 500 years could quote many thinkers from the ancient Greeks to young-earth creationists who lived around the year 2000, and based on this make the claim that the old-earth theory was a social construction. Would people then be right in believing this future hypothetical social constructivist? The answer is obviously no.

In another part of his article, Davies offers an interesting passage:

Platonism is relatively harmless, but no form does anything to explain why the orbits of the planets correspond so closely to the solutions of Newton’s law of
gravitation. Saying that the equations control the motion is vacuous unless one can at least begin to explain how this might happen, and I do not know of any significant
attempt to do this. My own approach is to admit that we do not know why the world exhibits so much regularity, but to regard this as a problem about the world rather
than about mathematics. Mathematics is simply our way of describing the regularity.

Contrary to Davies' statement, Platonism does indeed offer (the only available) explanation for the quantifiable (meaning it can be expressed quantitatively) regularity of the universe. It is however not an physicalist-positivist explanation of the kind that would satisfy Davies, and so he prefers to leave the whole question as a mystery. If our goal is to explain the universe, then it is better to take existing explanations seriously rather than adhering to some preconceived ontological commitments (i.e., physicalism) and hence ignore the explanations offered.
Davies quotes one of the most prominent living mathematicians, Michael Atiyah, who squarely puts himself in the conceptualist camp in saying that "Mathematics is part of the human mind". Of course there are several mathematicians who are Platonists, and several who are not (although they are in minority), but those who are not, like Atiyah and Davis, must leave the regularity of the universe and the unreasonable effectiveness of mathematics, as a mystical unexplained problem.

Vibrant Forms

The title of the blog comes from the following techno releases:

Fluxion - Vibrant Forms I & II

http://basicchannel.com/item/CRD-07
http://basicchannel.com/item/CRD-11

These organic abstractions (organic because they are structured and feel natural, unlike much abstract or minimal art) are perfect musical ways to the same thing I am trying to approach trough rational discussion on this blog.
Here is a track to listen to:

http://www.youtube.com/watch?v=UEWRnOjYuxk&feature=related

Comments on Mazur: Mathematical Platonism and its Opposites

The renowned number theorist Barry Mazur contributed an essay in the EMS Newsletter sequel on platonism. His article was published in the June 2008 issue, and should be taken seriously since Mazur is a first-rank mathematician, and as such has a singular insight into the experience of mathematical objects and facts.
Mazur positions himself securely outside both the platonist and non-platonist camps, but is clear that he thinks that any adequate philosophy of mathematics must take into account the deeply perceived experience of working mathematicians. Regarding platonism, Mazur views it as a kind of irrationalism, akin to blind faith:

If we adopt the Platonic view that mathematics is dis-
covered, we are suddenly in surprising territory, for this
is a full-fledged theistic position. Not that it necessarily
posits a god, but rather that its stance is such that the only
way one can adequately express one’s faith in it, the only
way one can hope to persuade others of its truth, is by
abandoning the arsenal of rationality, and relying on the
resources of the prophets.

Later on, Mazur advices platonists to learn from poets and prophets how to spread the faith in transcendent mathematical forms. This way of equating platonism with irrationalist religion is, I believe, a caricature of platonistic philosophy, and an oversimplification. Many writers have presented several different rigorous rational arguments for platonism, so it is not an area where we must completely abandon rationality. Have these rational arguments proved platonism? Certainly not, but neither have the opposites of platonism been proved. Such is the nature of philosophical questions, but just because they are not easily answered once and for all doesn't mean that we have to give up our own thinking and resort to prophets. Almost all philosophical positions are in the end a matter of faith, but it is still possible to argue rationally about philosophical questions, and thus base one's faith on a rational ground.

An interesting part of Mazur's essay consists of a list of advice for people who want to write about platonism versus non-platonism. As I am arguing for platonism, I will focus on what Mazur has to say to platonist writers. In essence, Mazur rightly observes that platonism implies a sort of disregard for rigorous proof. This is because platonists believe that mathematical facts are true irrespective of whether we can prove them or not, and that these truths are part of a platonist 'landscape' which it is our task to map. The challenge for the platonist, although Mazur does not say it explicitly, is thus to account for the demand for rigorous proof in mathematics while still maintaining that this demand is not strictly necessary. I would like to sketch an answer to this:

The method of rigorous proof is a tradition and social construction if there ever was one. The greatest minds of mathematics, from Euclid (who promoted the rigorous method, but nevertheless started his book with a proposition with a non-rigorous proof), through to Euler, Leibniz, and Newton, to modern day (not contemporary) Italian algebraic geometers and mathematical physicists (think Dirac's delta function and Feynmann's path integral) have worked on mathematics in a non-rigorous way. Clearly mathematics can be done in a not necessarily rigorous way, and fruitfully so! The insistence on rigour was, I guess, born some time around the time of Gauss and Abel, and developed in the hands of Weierstrass and others. Present day mathematics works in this tradition, even though one can observe cultural variations, e.g. in Russian mathematical exposition. Rigour has an 'hygienic' advantage in minimizing the number of incorrect statements in the literature, but it also no doubt slows down mathematical discovery. Where would Euler and Leibniz have been today if they had insisted on working rigorously?
Mathematics was discovered or created long before the idea of rigour, and it is not impossible that we may one day see the resurgence of not-necessarily rigorous mathematics. This has indeed been suggested, for example in the famous debate initiated by Jaffe and Quinn.

This is however not the end of the story. The platonist can argue that rigour, while not strictly necessary in the discovery of truth, is nevertheless our best tool for pinning down the exact nature of the mathematical forms, and so is valuable in that it gives us more exact knowledge.

Mazur demands that those in opposition to platonism must thoroughly account for the mathematician's perception of the transcendence and independence ("autonomy even") of mathematical concepts. I have written exactly the same thing in earlier posts on this blog. Whether non-platonists can ever convince us that we are in a deep illusion induced by spending much time thinking about abstract objects, and to lay bare some psychological mechanisms through which this illusion comes about, only time will tell. To me, it seems like a long shot. Until then those agreeing with Mazur have much reason to believe in mathematical platonism.

The non-mathematicians' common sense view, and why it is inadequate

There is a view of the nature of mathematics which is very common among non-mathematicians, and which, I think, by its very straightforwardness warrants the name 'the non-mathematicians' common sense view'. This view consists of the idea that mathematics is basically a language of abstractions of physical-world objects. The reasoning goes: First we counted fingers and pebbles, then we created an abstract idea of number that encompassed both, and from there mathematics developed. In this view, mathematics is no more than a language (albeit a formal one) to speak about abstractions of the real world, abstractions which are the products of human minds.

One advantage of this view, it seems at first glance, is that it gives an explanation to why mathematics is so successful in its applications to science. The common sense view would say that mathematics allows us to understand the physical world because it has been designed to do so. The claim is that mathematics is an organizational language constructed originally as a tool to deal with quantitative data and patterns observed empirically.
The first mistake that many non-mathematicians make is to think that mathematics is nothing but the language of the quantitative, or that which can be expressed numerically. Aside from the fact that this is more a description of statistics than of pure mathematics, this is a grosse misunderstanding of mathematics, but a misunderstanding that it takes a deep aquaintance of mathematics to spot. As a matter of fact, mathematics is much more 'qualitative' in its nature, dealing much more with concepts and structural relations, than the quantitative common sense view admits.
Practicing mathematicians who are generally well-aquainted with mathematics know that there is much more to mathematics than the language of quantity or abstractions of the physical-world. Why is this so? While it is no doubt true that the sequence of natural numbers 1, 2, 3, ... is an abstraction of physical phenomena such as counting pebbles or fingers, to say that all of mathematics is like this, is akin to saying that the music of Mozart is just a study in sound and a mental construction which comes directly from the natural sounds of insects, the wind, or other physical phenomena. Mathematics is, apart from a study of quantity, the science of precisely defined abstract structures. Some of the abstract structures which are fundamental to modern mathematics, such as groups, rings, or algebraic varieties, could (by a rather large stretch of imagination) be seen as abstractions and generalizations of the natural numbers or physical shapes such as curve-like objects. On the other hand, some of the most important objects in mathematics do not seem to be traceable back to counting fingers or to observing shapes of objects. Take for example group representations, infinite sets, sheaves, adeles, derived categories, etc. There is simply no reasonable way in which these objects can be reduced to simpler concepts or objects, until we reach the conceps and objects of our physical world.
But wait a minute, hasn't logicism (Russell et al) shown that all of mathematics is reducible to a few simple axioms of logic or set theory? Well, logicism has shown that mathematics is expressible in a formal language a posteriori, that is, after mathematics has already been constructed or discovered by non-formal means. The formal systems in question are almost completely non-conceptual at a higher level, that is, while they are capable of formulating our propositions and proofs, they would probably never have led to the discovery of the mathematical objects and concepts in the first place, and the meaning we give to the definitions in the formal system comes from the meaning of objects we have discovered previously. To claim that logicism proves mathematics to be an a priori study is thus outrageously unrealistic, because most of today's mathematics would never have seen the ligth of day if mathematicians had worked in the a priori setting of formal systems.
The reductionism of logicism is thus untenable, and the structure of the body of mathematics is better understood as a complex system with emerging properties. To be sure, the group representations could not exist without groups, and groups can be seen as vast generalisations of the set of integers, which itself is an abstraction and extension of finger counting. However, there is no immediate logically necessary path or physical metaphor between groups and group representations. The latter is simply an epi-entity of the former, something which could not exist on the simple level of the sequence of natural numbers itself, but that can be defined on a higher level, and indeed has profound implications for the objects on other levels as well as for physics, and even chemistry.
It is also a fact that many mathematical objects were discovered before (without any physical-world abstractions) their applications to physical phenomena. In higher science (i.e., way beyond finger counting), it is thus often a matter of concepts of mathematics informing empirical sciences, rather than the other way around.

The above discussion has made the point that mathematics cannot simply be an abstraction of physical objects and phenomena, as Aristotle thought. Aristotle would not allow infinite sets, and he would probably not have accepted such fundamental objects as negative numbers, not to mention complex numbers, which have no interpretation in the classical Greek geometry. Nevertheless, these mathematical objects are indispensable in today's science.

Now, if the objects of mathematics were just free creations of the mind, building on physical-world abstractions, then we would not be able to explain how these objects are so incredibly well-suited to understanding the physical world. Group representation theory was originally developped without any physical applications in mind. The physical study of atoms and elementary particles started independently of abstract algebra. Still, the two converged and came together in the application of representation theory to elementary particle theory. Moreover, this is not the only example of its kind. It is thus wholly unreasonable to assume that mathematics is a free game of the mind because we know that most abstract free games of the mind, such as many philosophical or political theories, card games, or board games, are useless for understanding the physical universe. Chess is an abstraction of objects in society (a kingdom), and it's played according to exact logical rules, much like mathematics. Still, chess is completely useless in explaining the universe, while mathematics is profoundly enlightening. In explaining the nature of mathematics one must therefore account for this difference in applictations to the understanding of the physical world. The success of mathematics in this area can, according to me, only be adequately explained if one understands the objects and results of mathematics as actually being facts of the world. But mathematics is not exactly facts about the physical world, because there are no perfect circles or infinite sets in the physical world. Rather, mathematics must be about general abstract facts of which the physical world is an instantiation, or particular manifestation.

Confessiones

If one is preoccupied with arguing for a thesis, ideology, or cause, one should from time to time take a step back and reflect humbly on what one is doing, and why. At the moment I am obviously convinced of the truth of the claims I present on this blog, and I believe there are good reasons and arguments behind the claims. Nevertheless, I hereby confess that:

It is possible that I am wrong about mathematical platonism.

It is possible that the people who reject mathematical platonism are right, and perhaps one day I will realise this. Still, I believe that mathematical platonism is a correct description of reality, and I find the arguments for it, together with the experience of many mathematicians, overwhelmingly convincing.

To confess that one might be wrong is an essential step in the pursuit of knowledge and intellectual development. Many people do not dare or cannot afford the luxury of admitting that they may be wrong, since their careers may depend on it. Most of Western society frowns upon people who admit the possibility of their opinions being flawed. Most politicians will not admit that they are wrong even if being proved so! In such a world, truth has taken a back-seat, and pragmata is king.
I am fortunate to be in the position where I can admit I may be wrong. This is partly because my subject matter is other-worldly; still, I believe it is not irrelevant to people's lives.