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
Friday, 19 June 2009
Thursday, 18 June 2009
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:
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.
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-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.
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.
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.
Thursday, 11 June 2009
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.
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.
Monday, 8 June 2009
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.
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.
Sunday, 7 June 2009
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:
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:
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.
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 theIf 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.
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 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.
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