Heisenberg's platonism

W. Heisenberg was one of the great physicists of the 20th century, and helped shape modern quantum mechanics. Around two thousand years after Plato and Aristotle, and even before the more recent developments of mathematical physics such as string theory, a leader of modern science has this to say:

I think that modern physics has definitely decided in favor of Plato. In fact the smallest units of matter are not physical objects in the ordinary sense; they are forms, ideas which can be expressed unambiguously only in mathematical language.

Hardy's platonism

G. H. Hardy was a famous number theorist in Cambridge, and the mentor of Srinivasa Ramanujan. He wrote a short book called A Mathematician's Apology, explaining his view of mathematics. It is a rather cynical book, devoid of any 'romanticism' about the mathematical profession. Hardy was sceptical of religion, and probably of spirituality in general. Nevertheless, he was a fully-fledged Platonist:

I believe that mathematical reality lies outside of us, and that our function is to discover, or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations' are simply notes on our observations.

Poem

See skulking Truth to her old cavern fled
Mountains of Casuistry heap'd o'er her head!
Philosophy, that lean'd on Heav'n before
Shrinks to her second cause, and is no more.
Physic of Metaphysic begs defence
And Metaphysic calls for aid on Sense!
See Mystery to Mathematics fly!

Alexander Pope, (1688-1744)

Manin's platonism

In the latest issue of the Notices of the American Mathematical Society, one can find an interview with eminent mathematician Yuri Manin, who is very highly regarded in the fields of algebraic geometry, number theory, and mathematical physics. He is also a mathematician who possesses a great breadth of cultural and historical knowledge.
Philosophers sometimes criticize mathematicians who express platonist leanings for being philosophically naive, or for not being aware of the latest developments in thought or empirical science. Examples can be found in earlier posts on this blog, where young Fields medalists were rebuked for expressing platonist opinions. In this way the huge number of platonist mathematicians are arrogantly dismissed as not having reflected on their subject maturely and according to current philosophical standards. This is definitely something that Manin cannot be accused of.

The interview reveals that Manin is not afraid of using the 'P'-word (my italics):

What has changed in pure mathematics?
The unique possibility of doing large-scale physi-
cal experiments in mental reality arose. We can
try the most improbable things. More exactly, not
the most improbable things, but things that Euler
could do even without a computer. Gauss could
also do them. But now, what Euler and Gauss could
do, any mathematician can do, sitting at his desk.
So if he doesn’t have the imagination to distin-
guish some features of this Platonic reality, he can
experiment. If some bright idea occurs to him
that something is equal to something else, he can
sit and sit and compute a value, a second value,
a third, a millionth. Not only that. People have
now emerged who have mathematical minds, but
are computer oriented. More precisely, these
sorts of people were around earlier, but, with-
out computers, somehow something was missing.
In a sense, Euler was like that, to the extent that
he was just a mathematician—he was much more
than just a mathematician—but Euler the math-
ematician would have taken to computers passion-
ately. And also Ramanujan, a person who didn’t
even really know mathematics. Or, for instance, my
colleague here at the institute, Don Zagier. He has
a natural and great mathematical mind, which is at
the same time ideally suited to work with comput-
ers. Computers help him study this Platonic reality,
and, I might add, quite effectively.

At this stage it is not clear what Manin believes regarding platonism. He refers to the mathematical universe as both a "mental reality" and (twice) as a "Platonic reality". This mystery of Manin's opinions deepens even further a little later on, when he says

I am an emotional Platonist (not a rational one: there are no rational
arguments in favor of Platonism). Somehow or other, for me mathematical research is a discovery, not an invention.
I imagine for myself a great castle, or something like that, and you gradually start seeing its contours through the deep mist, and begin to investigate something. How you formulate what it is you’ve seen depends on your type of thinking and on the scale of what you have seen, and on the social circumstances around you,
and so on.

This is puzzling for several reasons. First of all, to say that there are no rational arguments of platonism is not true in the most reasonable and common interpretations of these words. There are about as many rational arguments for platonism as there are for most other philosophical opinions. What Manin must have had in mind is the fact that platonism cannot be fully described in terms of today's science (which works in the paradigm of space-time and materialism).
Nevertheless, Manin seems to truly believe that mathematical platonism is a fact since he clearly states that he believes mathematics is discovered, not invented. His belief is reaffirmed in his platonist confession:

Gelfand: Are these properties of the problems
themselves, or is it just that no one is actively inves-
tigating them, for some social reason?
Manin: As a Platonist, I know that this is a
property of the problems themselves, but it is a
property that one cannot recognize at the moment
of formulating the problem. It reveals itself in the
process of historical development.
Partly for this reason, I am not partial to prob-
lems. Solving a problem requires the skill of
finding a detail, but you don’t know what it is a
detail of. As a Platonist, I am partial to complete
programs. A program arises when a great math-
ematical mind sees something as a whole, or not
as a whole, but as something more than a single
detail. But it is seen at first only vaguely.

The conclusion is that Manin has deeply held platonistic beliefs, but admits he has no scientific explanation for how the mathematical realm can exist in the way posited by platonism.

Pythagoras

The vision of Pythagoras 2500 years ago that number, shape, and relation underlie reality and pervade human knowledge, is in light of the tremendous success of mathematical methods in science and the digitilization of music, one of the most successful predictions made in human history.

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.

Reply to Hersh: On Platonism

Reuben Hersh is a well-known contributor to the debate on the philosophy of mathematics. He wrote a follow-up to Davies' article, also published in the EMS Newsletter (June 08). He seems to be in general agreement with Davies, but focuses on a different aspect of criticism of platonism, namely the social-construction view of mathematics.
Already in the first paragraph, Hersh declares that most practicing mathematicians have a "rough-and-ready, naive" view of the philosophical underpinnings of mathematics. This is a very typical comment which mathematicians have heard from philosophers so many times that they finally stopped listening. It is a rather arrogant attitude to claim that people who spend a lot of time reflecting on various arguments for or against ideas, have a more developed and mature view of things than people who deal with the actual objects in question on a day to day basis. Since Hersh has written much about the importance of founding philosophy on the experience of mathematics, one might have hoped that he would pay specific attention to the experience of mathematicians who de facto experience mathematics closer and deeper than anybody else. Unfortunately, Hersh has chosen the view that most mathematicians do not understand what it is they are working with, and are philosophically naive and illusioned.
This type of attitude of philosophers towards mathematicians has been very damaging for the relations between philosophy and mathematics in the past, and one may hope that the future of the philosophy of mathematics will be more interested in a dialogue with mathematicians, rather than continuing the tradition of dismissing mathematicians as philosophically naive.

Hersh chooses to reject Platonism on the grounds that it is incompatible with the materialist outlook of reality, a view whose correctness he takes as self-evident and unquestionable. This type of view is a relic from the positivist era, and as such has lost much of its power since the days when it was at its most popular. Nowadays there are many rational scientifically-minded people who are not materialists, and we now know that materialism/mechanism/reductionism and so forth are not proved by science, but in fact just assumptions that one may or may not hold in connection with a scientific view of the world. One simply can't base an argument against platonism on a premise that lacks any proof or self-evident qualities.
One problem with Hersh's social-construction view of mathematical reality is that it could just as well be applied to, for example, physical science. If mathematical objects are social constructions, then why are not the objects of physics, such as planets, forces, energy, also social constructions? Despite the social aspects of science, its 'corporeal ground', its postmodern critique, deconstruction, and whatever, almost no sane person will deny that planets are really 'out there'. In the same way, a social-construction view of mathematics cannot necessarily imply that the platonist 'out there' view must be rejected. Of course there are cultural aspects to any human activity, but most people still hold that there are matters of fact beyond the cultural sphere. These facts are relations between objects existing independently of human culture. What this comes down to is the much repeated observation that a social-construction view of mathematics can't be used as an argument against platonism, it can only be used as an alternative after one has chosen to reject platonism.
The article also mentions the problem of mathematical 'intuition' and how, if the platonic realm exists, humans can obtain knowledge of it. Here Hersh mentions the curiously irrelevant observation that blood flow through various parts of the brain can be correlated to mathematical thought. As I have written before in the post on Davies' article, this is completely irrelevant as to the existence of a platonic realm, but again, it can be used to fill the vacuum after one has chosen to reject platonism. Hersh completely agrees with this when he writes:

From this point of view, the facts presented by Davies are relevant and interesting. They do not refute Platonism, they are part of the scientific program that one focuses on after rejecting Platonism.

In spite of Hersh being (rightfully) sceptical of Davies' line of argument in rejecting platonism, Hersh agrees with Davies in dismissing the possibility of a mathematical intuition interacting with a platonic realm:

It seems that Davies regards the evidence that thinking takes place in the brain as proof that there is no such “intuition”, in the sense of a special mental faculty for connecting to “out there.” But with or without neurophysiological evidence, it is pretty clear that the posited “intuition” is an ad hoc artifact, lacking any specificity or clear description, let alone empirical evidence.

The lack of specificity or clear description of the interaction between the mind and the abstract objects of platonism is no argument for or against the existence of such a function of the brain.
As I described in the post on Davies' article, we have no specific or clear description of how memory is stored, or how consciousness interacts with it or with other faculties of the mind. Does this provide a basis for rejecting the existence of stored memories in the mind? Obviously not. It is probably a risky and possibly not very useful strategy to limit one's philosophical positions to what science can tell us at the moment. Philosophy is at its most fruitful when it is not pretending to be science.
Hersh formulates his materialistically based conclusion despite admitting the lack of the proper empirical evidence asked for above:

The mental, social and cultural, including the mathematical, are grounded in the
physical – the flesh and blood of past and present humans, especially mathematicians. We can recognize this, even while the detailed nature of this grounding – just how our thoughts are carried out by our brains – may
never be completely understood.

If one has chosen ontological materialism, then Hersh's arguments are a natural and rational way to proceed from there. However, this position has some major disadvantages. Firstly, as we have already observed, it demands that one ignores the overwhelming consensus and experience of practicing mathematicians, and therefore alienates a whole group whose opinion is most relevant for the matter. Secondly, one looses the possibility of a good explanation of how mathematics can be so universal, stable in time, and above all, utterly useful for understanding the physical universe. Hersh dismissively calls this a puzzle (no doubt, he chose this word consciously, rather than the word 'problem'), while in fact it is a major problem and a great mystery (to use Wigner's words), a mystery whose only satisfactory explanation to date is platonism. It is nowhere nearly enough to say that all human cultures can observe and count pebbles or fingers, and that therefore they develop the same mathematics, and as long as we can see pebbles and count our fingers, mathematics will be universal. The thing is that the universality and timelessness of mathematics is manifest on a much higher level which cannot simply be reduced to counting pebbles. Examples of this include the remarkably congruent and simultaneous but independent discoveries of infinitesimal calculus (Leibniz and Newton), and non-Euclidean geometry (Lobachevski, Gauss, Bolyai), respectively. An example of the 'mysterious' effectiveness of mathematics is the application of group theory (which started as an area of pure mathematics) to physics.
If one adheres to platonism one must explain how our minds interact with the platonic realm. If one rejects platonism one must explain why mathematics is culturally and geographically universal, tolerant to the test of time, and why it is that mathematics is so efficient in describing our physical universe, even when it was not originally conceived for this purpose! These two are both very difficult problems, but given that we go beyond scientism and materialism, we have a chance of finding good solutions to them. I claim that reality does not simply consist of things that are empirical on the one hand, and things which are socially constructed on the other. This type of ontology (empirical objects + social constructions) is of course what remains after the two big trends in Western thought, exemplified by positivism and social constructivism, respectively. The problem is that both of these two trends ignored, or did not provide an adequate place for, a fundamental domain of intellectual pursuit: mathematics.

The problem of interaction

The main argument against platonism is the problem of how we as human beings can interact with and obtain knowledge of, the platonic realm. In some ways one might say that arguments for platonism, by way of persuasiveness, stand or fall with the way in which they deal with this problem.
As I see it, the problem lies in the way in which we intuitively imagine the meaning of interactions. After all, a philosophical argument, given that it is logically correct, will be persuasive exactly if it is intuitively appealing to a majority.
Despite the modern advances in physics, most people probably still imagine interaction between two entities as two completely separate bodies, and some point in which they touch, or with some ray of energy going between them. This picture of interaction is the psychological source of the problems with dualism and platonism alike, because we can't have any direct physical touch-contact with things that are non-material, and moreover, we don't observe any "rays of energy" or anything similar going from somewhere outside the universe into our brains. So how can humans interact with the platonic realm? This question is very deep and difficult, so we can only begin to sketch an answer, which is by no means complete at this stage.

As stated above, the salient point is the ways in which we can model interaction. We have seen that the Newtonian billiard-ball model of interaction won't do, and neither will the more modern wave-mechanistic paradigm. After all, numbers do not hit us the way balls do, and neither do they induce processes in our brains after being picked up, as waves, by our ears or eyes. What other ways could there possibly be in which to model interaction? The answer may come if we consider analogies with more concrete phenomena. One example is:

How does memory interact with consciousness?

We know that memory is 'dislocated' in the brain, that is, there is no specific parts of the brain that carry memories. Still consciousness clearly interacts with at least some of our memories. We currently do not understand how this works, but whatever the mechanism, it seems very likely that it must be an interaction of a type very different from our everyday life intuitions. It is clear that consciousness and memory are partially separate entities, in the sense that we can be conscious without remembering the names of all the capitals of the world, and conversely we have memories that we are not immediately conscious of. Still these two entities must be considered as somehow intertwined, because consciousness is not physically touching a new point when it digs up old memories. The point is that it is not necessarily a question of an interaction that takes place as a transposition in space. We tend to think about interactions as points in space such that before the interaction there were two separate bodies, and during the interaction there is at least one point in space where the two entities meet. With our present understanding of consciousness and memory, this touching-point model seems to be inadequate.
I do not pretend to be able to explain how consciousness interacts with memory, but I claim that this phenomenon points to a new way of understanding interactions between entities, specifically between 'dislocated' entities.
It is naive and misleading to think about the platonic realm of mathematical entities as somehow 'outside' our physical universe. In the same way it would be wrong to think about memory as spatially located outside of consciousness. Still, not all our memories are in our consciousness, and indeed, at least some of our memories exist independently of our wanting them to be there, and independently of whether we are currently conscious of them or not.

This is an important point which I should follow up another time. Hopefully there will be more to come on this subject.

Reply to Davies: Let Platonism die

It's time to confront the series of articles on platonism published in recent issues of the EMS Newsletter. The first one that was published was E. B. Davies' article entitled Let Platonism die. Inclusion of words derived from death or dying usually indicates some zealous semi-political agitation, but let us instead focus on the arguments Davies puts forward:

Davies starts by expressing indignation over the fact that several Fields medalists have spoken positively about their platonist convictions. After quickly dismissing mystical experience as nonsense, in the best style of scientism or positivism, Davies seems to jump to conclusions when he states:

Although he is a Platonist, Roger Penrose is almost
unique in accepting that his beliefs imply that the math-
ematical brain cannot obey the known laws of physics.

It is true that Penrose is a platonist. However, it does not follow from platonism that the "mathematical brain" cannot obey the known laws of physics. Why not? Well, even though platonism postulates a type of perception with the capacity to reach beyond the physical world as currently understood, this does not imply that the mechanisms of the brain must themselves go beyond the laws of physics as currently understood. This only follows if one also assumes that the universe is causally closed, that is, that something beyond the physical universe cannot interact with the physical universe. This is something which platonism does not assert.
There are many aspects of the human mind that are currently not understood. Indeed, nobody would claim that we understand for example how human consciousness works. Platonism implies that there is a brain function which is not covered by today's explanations of the workings of the mind. This is not a very radical claim, because there are many brain functions which are currently not understood. If you claim that consciousness exists as a brain function, then you are claiming that the human brain works in ways which are not explained by current science. Does this make you an unscientific mystic? Obviously not. Does this imply that consciousness must be due to some physical laws not currently known? No. If one reads Penrose one learns that his original motivation to develop his ideas about consciousness as a product of quantum effects was to construct an alternative to the predominant view of the mind as a computer. This view of the brain as a computer does not fit well with Gödel's incompleteness theorem, and this, rather than any necessary need to defend platonism by postulating new physical laws, has been Penrose's goal.

Given the failure of Davies' claim that platonism requires some extra-physical brain mechanism, the remainder of his argument looses its relevance. Davies mentions how modern brain science has revealed mechanisms of sense perception and the perception of number. This is all very well, because of course we understand mathematics with our brains, and of course the way in which we understand certain things can be described in terms of neuro-cognitive models. Platonists would not deny such obvious facts. The point is that it is fully possible that our brains are able to produce mental states of awareness of objects whose existence is not dependent on the spacio-temporal world, but which nevertheless interact with it because these objects provide the forms in which the landscape of the universe takes shape.

Two Fields medalists on platonism

The Fields medal is the most prestigious prize that is awarded to mathematicians. One of the most recent Fields medalists, Andrei Okounkov was interviewed in the EMS Newsletter. Among other things, he didn't hesitate to express the following in reply to a question:

Much of your work has deep connections to physics. Does
that mean that you find it essential that mathematics is
related to the natural word, or that you would even think
of it as subservient to the other natural sciences?

O: When I said “our world” earlier I didn’t mean just the
tangible objects of our everyday experience. Primes are
as real as planets. Or, in the present context, should I say
that celestial bodies are as real as primes?

What I hope will slowly emerge from various posts on this blog is that a large majority of the best mathematical minds see mathematical platonism as a natural and convincing position. The skepticism often found among philosophers and empirical scientists does not take this fact seriously. Either mathematical platonism is true, or mathematics is the only area of human intellectual endeavour where its most prominent practitioners have less understanding about the nature of their subject than people from outside that area.
Here is an excerpt from the same article; an interview with Terrence Tao, who won a Fields medal the same year as Okounkov:

Many mathematicians are Platonists, although many may
not be aware of it, and others would be reluctant to admit
it. A more “sophisticated” approach is to claim that it is
just a formal game. Where do you stand on this issue?

T: I suppose I am both a formalist and a Platonist. On the
one hand, mathematics is one of the best ways we know
to try to formalise thinking and understanding of con-
cepts and phenomena. Ideally we want to deal with these
concepts and phenomena directly, but this takes a lot of
insight and mental training. The purpose of formalism in
mathematics, I think, is to discipline one’s mind (and fil-
ter out bad or unreliable intuition) to the point where
one can approach this ideal. On the other hand, I feel the
formalist approach is a good way to reach the Platonic
ideal. Of course, other ways of discovering mathematics,
such as heuristic or intuitive reasoning, are also impor-
tant, though without the rigour of formalism they are too
unreliable to be useful by themselves.

Two remarks are in order here. First of all, the interviewer obviously didn't find a better word to use than "sophisticated". One must not be led to believe that formalism is somehow deeper, better developed in its details, or somehow the preferred view of sophisticated minds. What was probably meant above was that platonism is the "first" position one naturally develops as a mathematician, and that with later reflection, one might be led to formalism. Obviously this doesn't say anything about the relative merits of truths of the two positions.
Secondly, Tao says he's both a formalist and a platonist. The common meaning of these terms, as used in the philosophical debate, is such that one excludes the other. Mathematical formalism as a philosophical position usually means that mathematics is ontologically seen as formal symbols. Platonism on the other hand posits the independent existence of mathematical objects, beyond the mere symbols, which are modelled by the formal systems used in mathematics. The latter seems to be the position of Tao, and it's a fair bet that most working mathematicians (but not all, of course) would agree.

The common perception of mathematics

I have from time to time brought up the question of the nature of mathematics in discussions with people from other disciplines. The common perception of mathematics by non-mathematicians is that mathematics is simply the language of the quantitative, a formalism developed to solve real-life problems. It follows from this view that mathematical objects have about the same ontological status as the words of normal language - at best they point to something that's real, namely, physical objects, but they are not themselves real. It is difficult to explain to non-mathematicians why this view is an inadequate description of mathematics. In the June 2000 issue of the EMS Newsletter, Jean-Pierre Bourguignon writes:

Some fallacies about mathematics. For
some people, mathematics is just the language
of the quantitative. They base their judge-
ment on the fact that mathematics enter-
tains a special relationship with language;
this opinion is shared by some of our fellow
scientists. We mathematicians know how
wrong this is, and how much effort goes
into building concepts, establishing facts,
and following avenues we once thought
plausible but turn out to be dead ends.
This widespread belief forces us to consid-
er more carefully how mathematics inter-
acts with other disciplines and what is the
exact nature of the mathematical models
that appear ever more frequently.

The challenge of explaining to non-mathematicians the perception of mathematics obtained from years of deep study of the subject is made difficult for several reasons. There seems to be a general mistrust, in academic or philosophical discussions, of the judgement of mathematicians regarding their own subject. After all, wouldn't a long and intense study of the characters of the Lord of the Rings eventually engender the view that these characters have an existence of their own?
There is a cognitive difficulty in understanding mathematics the way working mathematicians understand it. Not only does one need a tremendous effort of abstraction, it is also not enough to just read about the results of mathematics in reviews or books, because these presentations are usually mathematics after it has crystallized into formalism. Mathematics as a living development is a completely different thing (as mentioned in the above quote), and this can only be understood by actually engaging in research, or by listening to what mathematics have to say about their work. Here is Bourguignon again:

A special link with truth: Confronted with
a mathematical statement, a mathemati-
cian's goal is to prove that it is "true". What
does this mean? Of course, now that math-
ematicians agree to work in the context of
a potentially completely formalised theory,
ts meaning can be none other than 'the
statement can be deduced from the basic
axioms agreed upon'. In fact, if we are to
address this question in the context of the
relations of mathematics to society, we are
forced to take it in a broader, more philo-
sophical, perspective because we have to
confront it with reality. This amounts to
deciding whether there is a mathematical
reality of which this statement is a part,
and of the status of this reality in relation
with ordinary 'sensible' reality. On this
broader issue, mathematicians have differ-
ent views: from the 'Platonists' at one end
of the spectrum who believe that they 'dis-
cover' new territories and new facts of the
mathematical world, to the 'intuitionists' at
the other end who view mathematical con-
structions as purely human scaffoldings
based on consensus opinions among a
restricted community - in other words,
that mathematics is 'invented'. I have no
doubt that the vast majority of mathemati-
cians are closer to the Platonist viewpoint
than to the other one, or at least that they
spend a good portion of their professional
time behaving as if they were, as André
Weil once put it.

One of the symptoms of non-mathematicians perception of mathematics is the view that mathematics is not a science. If maths is a formal language that deals with logical relations between fictional entities or mental constructions, then how can it be a science? Many scientists view science as an empirical activity, first and foremost. For most working mathematicians this is an unfamiliar view which does not resonate with their everyday experience of mathematics as an activity of uncovering synthetic and deeply meaningful truths. In his conclusion Bourguignon writes:

From all the above challenges concerning
the image of mathematics, the most serious
seems to me the need to make it evident that
mathematics is a science and alive. The rest
should follow.

Evolution and human nature

The Templeton Foundation has asked several distinguished researchers the question

"Does evolution explain human nature?"

The responses vary, and are in the form of essays available here:

http://www.templeton.org/evolution/

Of particular interest is Martin Nowak's reply which mirrors the ethos of this blog in a sharp and succinct way. Nowak is professor of biology and mathematics at Harvard University, and has this to say:

"Music is part of human nature. There is also something very intuitive about numbers and geometric objects, and the ability to do some basic math seems to be part of human nature.

Yet the great theorems of mathematics are statements of an eternal truth that comes from another world, a world that seems to be entirely independent of the particular trajectory that biological evolution has taken on earth. The great symphonies of Beethoven and Mahler capture glimpses of a beauty that is absolute and everlasting. Beyond the temporal, materialistic world there is an unchanging reality.

My position is very simple. Evolution has led to a human brain that can gain access to a Platonic world of forms and ideas. This world is eternal and not the product of evolution, but it does affect human nature deeply. Therefore evolution cannot possibly explain all aspects of human nature."

Grothendieck's platonism

Alexandre Grothendieck is one of the last century's greatest mathematicians. There is no controversy in describing him as the mathematician who has pursued the most abstract type of mathematics ever. In his mathematics there are usually no numbers directly involved. Most of the time there are even no letters denoting numbers involved. The level of abstraction is breathtaking and, to some, deeply fascinating. Nevertheless, his work does have more concrete implications for relations between numbers. There is a huge collection of articles by and about him on the internet. A brief recent introduction can be found here:

http://www.sciencenews.org/view/generic/id/31898/titl/Math_Trek__Sensitivity_to_the_harmony_of_things

The following passage from Grothendieck's Recoltes et Semailles expresses, in a somewhat poetical way, his scientific realism, in particular mathematical realism. It seems like he is saying that what we call inventions and imagination are in fact very close to what platonists refer to as intuition or 'the mind's eye':

"What makes the quality of a researcher’s inventiveness and imagination is the quality of his attention to hearing the voices of things."

Perhaps Grothendieck is also saying that not only is the platonic realm a static life-less structure which we can tap for information, but he seems to suggest that physical and metaphysical entities are actively revealing themselves in various ways, ways which we of course may or may not pick up.

Marcel Légaut

A. Grothendieck wrote a list of a number of people he called "Mutants", that is, people who in his view were spiritually ahead of their time. I just looked up a name from his list which I had never heard before:

http://en.wikipedia.org/wiki/Marcel_L%C3%A9gaut

This seems like a very interesting personality. I hope I can find some good source about his life and thought. The above Wikipedia article is unfortunately written by someone who is not particularly interested in producing texts where every word carries a meaning which can be picked up by a potential reader.

Naming Infinity - maths and spirituality

There is a book that's still hot off the press, called:

Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity

The authors, Graham and Kantor, tell the story of the rise of the famous Russian school of mathematics, and it's intimate relationship with an Orthodox movement (later deemed heretical) called 'name worshipping'.
One of the main interesting features about this work is the way in which it illustrates, by way of historical description, how mathematics and religious spirituality can still, thousands of years after Pythagoras, inform each other in a practically efficient and fruitful way.

Russell said that mathematics is the chief source of the belief in exact and eternal truth. Of course he had a priori, due to his other beliefs, discarded another important source of such belief: religion, or more generally, spirituality. What is the unerlying reason why these two areas of human endeavor keep popping up hand in hand throughout human history? Howcome they are so seemingly different, yet clearly intimately connected? One answer is to explain everything in terms of human brain function and biological evolution. However, these kinds of explanations fall short of explaining the unchangeing characters (Russell used the word 'eternal') of these subjects. This, and other drawbacks of the "all in the mind" theory, is something I will return to in this blog.
Another explanation is given by a model which includes non-empirical and unchangeing objects which constitute the 'forms' of the physical, inherently dynamical and changeing, entities we observe empirically.

Freedom is to be able to say the Truth

In George Orwell's famous novel Nineteen Eighty-Four, the protagonist Winston Smith writes:

"Freedom is the freedom to say that two plus two makes four. If that is granted, all else follows."

The phrase "two plus two equals five" was used in communist USSR to refer to how five-year plans were supposed to increase production to yield more than previous methods of production. Everybody knows what this denial of truth led to. However, this scenario is not something exclusive to so-called totalitarian regimes. It also has other manifestations:

• The current financial crisis might at one level have its foundation in the (tacit) belief that money can grow on its own - that one can somehow get more than there actually is.
  
• Relativism is very popular in today's Western world, and ideas of social construction permeate contemporary thinking. In some parts of this view, science is just useful conventions and has no reality outside of human existence.
  

As Aldous Huxley anticipated already in the 1930s, the dictatorship is not likely to come the way George Orwell thought, that is, via forceful external pressure, but it will gradually grow into our lifestyle in ways which we welcome, because we find them pleasurable. However, in contrast to what both Orwell and Huxley thought, the dictatorship does not seem to be a regime. It's not a government who wants to control the citizens. It's our very culture which is going to force us into dead-end roads, like that of relativism and individualism.
As the examples above show, the way this can happen will most likely go via a rejection of objective truth. In this regard, the ultimate triumph is to overtake the main bastion of objective truth: mathematics.
To quote again from Nineteen Eighty-Four:

"In the end the Party would announce that two and two made five, and you would have to believe it. It was inevitable that they should make that claim sooner or later: the logic of their position demanded it. Not merely the validity of experience, but the very existence of external reality, was tacitly denied by their philosophy. The heresy of heresies was common sense. And what was terrifying was not that they would kill you for thinking otherwise, but that they might be right. For, after all, how do we know that two and two make four? Or that the force of gravity works? Or that the past is unchangeable? If both the past and the external world exist only in the mind, and if the mind itself is controllable—what then?"

Contemporary relativists think that they are pursuing a liberating ideology. It is more likely, given the evidence, that the result will be the exact opposite. Objective truth has always been a defender of freedom, and mathematics is its most clear and universal manifestation. Freedom is to be able to say that mathematics is true and unchangeable no matter what our brains are like, and no matter what our culture and politics says.

Relative plausability of platonism compared to standard hypotheses

Science shows that reality is so rich and complex that explanations will inevitably lead to scenarios which from our every day perspective seem fantastic, counter intuitive, and beyond common sense. Philosophy on the other hand, uses as one of its most widely applied tools of argumentation, the principle that a statement, in order to be believable, must be readily understandable, at hand, and well-grounded in the prevailing ontology (nowadays most often materialism). Science provides many examples that this principle is inadequate in our pursuit of understanding of the world. Indeed, there exist examples where scientific reasoning leads us to quite fantastic scenarios as the best explanations of the world.

Modern theoretical physics, in considering the anthropic principle, has brought us to basically two plausible scenarios: Either there is a Creator who has fine tuned the physics in our universe so that to make life like ours possible, or there is an incredibly high number of universes besides ours, where all logically possible physical theories are realized.
Now, which of the following would you say requires the greatest leap of faith, mathematical platonism, which posits a realm of mathematical objects and relations existing independently of our own minds, or the existence of God or infinitely many universes outside of our own?
If string theory with its ten spacial dimensions or the multiverse with billions (or even infinitely many) different universes outside our own are the best explanations for how physical reality is, and if it is reasonable for a rational person to believe in such theories, then what is so strange about the belief that mathematical objects exist independently of our minds?

It is a simple task to realize that both God and the structure of masses and masses of different universes are more complicated structures than the realm of mathematics. Still it is rational to believe in one of these extremely complicated transcendental existences. In comparison, a platonic realm of mathematics, seems like a small grain of sand.

The above examples show that the philosophical argument against mathematical platonism based on the difficulty to imagine where a transcendental world like the platonic realm could exist, is an argument which is not universally applicable. It appeals to down-to-earth bias (one could say geocentrism, to resurrect an old piece of terminology in a new context), but there exist situations where it is misguided, for example in the multiverse scenario mentioned above. To realize the shortcomings of the geocentric argument, one has to take a huge leap into abstraction. To cling to the argument is to effectively say "I can't make the required leap of abstraction, therefore I don't believe in the possibility of those things".

The most vocal current anti-platonists do not reject platonism mainly on account of the geocentric argument, but because they are in the business of promoting their own alternative views which underlie their research agendas. Did you expect anything else?

Musical platonism

Mozart is the greatest composer of all. Beethoven created his music, but the music of Mozart is of such purity and beauty that one feels he merely found it — that it has always existed as part of the inner beauty of the universe waiting to be revealed.

- Albert Einstein



Mozart's music is the mysterious language of a distant spiritual kingdom, whose marvelous accents echo in our inner being and arouse a higher, intensive life.

- E.T.A. Hoffmann

New Scientist article about mathematics and computing

A recent article in New Scientist

http://www.newscientist.com/article/mg20126971.800-rise-of-the-robogeeks.html

describes, in the typical vague and sensationalist terms characteristic of so-called popular science journalism, some work of computer scientists Aaron Sloman and Alison Pease, respectively.
After the initial spectacular suggestion that artificial mathematicians may become a reality, the article eventually admits that the actual aim of the research presented is much more modest:

"at this stage he [Sloman] is simply trying to show a link between spatial manipulations and the basics of mathematics."

One should be very careful in extrapolating this into believing fantastic stories about "robot mathematicians". It is also important to remember that the idea of mathematical machines that could prove theorems more or less independently of human input, is not a new one. What can be done today to some extent, is automatic proof check. Several mathematicians have constructed programs that can formally check the validity of certain non-trivial theorems in mathematics, that is, the program finds a deduction from the axioms to the full theorem, using only formal rules of logic. It is quite possible that similar methods could be used to find previously unknown proofs of mathematical results or conjectures, or to handle routine arguments that mathematicians today have to do by hand.
However, mathematics does not simply consist of formal truths deducted from logic, just as a painting does not simply consist of spots of paint on a canvas. Many formal truths are not relevant, and human mathematicians can easily exclude those, but it is very hard to imagine a computer program which would recognize this unformalized (unformalizable?) difference. For instance, a computer might come up with the theorem 2 + 2 = 4, which is relevant, but it could just as well come up with the equivalent theorem "the even prime number plus itself equals the smallest positive even integer square", which is not a mathematically relevant theorem. How is the computer to know which one to choose? It seems it would need human input.
It seems unlikely that a computer, which necessarily works with a formal system as a model of mathematics, will, from some rudimentary axioms alone, come up with such creative products of the human mind as integrals, the category of groups, or etale cohomology, concepts of central importance in modern mathematics. This is because although the proving of theorems seems to be formalizable, the formation of good mathematical definitions, that is, fruitful and enlightening concepts, is something that seems to require human judgement.

The black-and-white statement by Pease that mathematics is not discovered but created is just an empty slogan unless one provides some good argument for it. It is also something that many professional mathematicians would object to. (Indeed, comments at the New Scientist website by readers of the article react against precisely this statement).
It is striking that an article in New Scientist about the possibility of artificial mathematicians, do not feature any statement by an actual mathematician. If you want the best possible opinion about your health, you ask a medically trained person. If you want the most accurate description of history, you ask a historian. If you want the best insight into mathematics you ask a...computer scientist or cognitive psychologist? This reminds me of a quotation by A. N. Whitehead:

"Philosophers, when they have possessed a thorough knowledge of mathematics, have been among those who have enriched the science with some of its best ideas. On the other hand, it must be said that, with hardly an exception, all remarks on mathematics made by those philosophers who have possessed but a slight or hasty and late-acquired knowledge of it, are entirely worthless, being either trivial or wrong."

Chartres cathedral

Here I am standing
as if I was here already a thousand years ago
at this work of human hands
a gift to infinity and grace

Comment on Quine and the formalist project

I posted this comment on Facebook after I found at least two groups on Quine, and another on analytic philosophy. A paradigm shift is waiting around the corner and would be well underway already, were it not for the fact that most professional mathematicians today are uninterested in philosophy.

The comment:

Pragmatical justification of logical or mathematical truth is a misunderstanding that comes from the outdated formalist identification of these truths with truth values of propositions in formal systems. Euclid's mathematics was and is true for two thousand years, even though its various axiomatizations may vary in terms of correctness, completeness, consistency, etc.
Formal systems are not natural, they are artificial languages constructed in order to model the natural process of mathematical discovery. In contrast, many of the results in mathematics are very natural because thousands of years of empirical evidence has not falsified them.

Techno music illustrating the platonic realm

For those unfamiliar with techno or dub, Moritz von Oswald is one of the most influential and highly regarded producers of techno and (second wave) dub music. He forms one half of legendary labels such as Basic Channel, M (Maurizio), and Rhythm & Sound, and is well-known for almost never appearing on photos or in interviews...until lately.
In a recent lecture/interview, he talks about, among several other things, one of his (in my opinion) most profound pieces of dubtechno: M4.5. The most notable idea he put forward was that certain tracks have a quality about them which he described as music which is standing still. Although it is an inherent quality of techno music to be repetitious and sometimes monotonous, the quality of standing still is more apparent in some tracks than in others, and is not directly related to being monotonous. The M-series is indeed an evolving type of techno, although it does not evolve to any specific point, and can therefore not be said to be progressive. Still the tracks on the M-series induce an effect of a picture, rather than a movie.
It seems that what von Oswald is talking about is the possibility of eliminating the timeline from the experience of techno music. Of course this doesn't mean that the music is not moving forward as the track progresses, but that it induces an experience in the listener more like a fixed picture than a moving series of sounds. This opens up the fascinating possibility of techno music taking on more of the character of the unchanging objects of mathematics.