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mathematics is full of circular reasoning, infinite regress and paradoxes. It may be exist, it may be fictional.

Of course, it has no more infinite regress than any other discipline that depends on logic; but that the point, most don't see, it's seen as axiomatic, almost as Gods own truth, but more so; mathematical & logical facts have a status over other kinds of truth that perhaps they don't deserve.

Similarly with physics. I can recall just how shocked the students were when a professor seemed to posit a particular idea without justification, prior proof etc; it seemed like magical thinking to them. This was in an advanced class - quantum field theory. (He was the manifestly the best lecturer, gathering a sustained round of applause at the end of his course, an honour not accorded to any other lecturer by the student body).

Eugenia Cheng, a professional mathematician has said, that when she talks to philosophers, she's convinced that it doesn't exist, that it is problematic; but when she returns to her work, there they are, the concepts she handles, as solid as they can be before her minds eye.

Can professional mathematicians/physicists actually assimilate how problematic mathematics/physics is, if they are to do their work, if they are too see the concepts they handle with concrete clarity, rather than as misty illusions. Rather than liberating, it may be disillusioning. Like piercing the veil of maya.

note: Fictionalism is best thought of in opposition to Platonism, it "is the view that (a) our mathematical sentences and theories do purport to be about abstract mathematical objects, as Platonism suggests, but (b) there are no such things as abstract objects, and so (c) our mathematical theories are not true. Thus, the idea is that sentences like ‘3 is prime’ are false, or untrue, for the same reason that, say, ‘The tooth fairy is generous’ is false or untrue—because just as there is no such person as the tooth fairy, so too there is no such thing as the number 3"

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Would you be willing to rephrase some of your question? Speaking as a mathematician, I can tell you that there's no more infinite regress than any other enterprise which relies on logic, that circular reasoning is not generally accepted, and that in any case mathematics exists. Are you not embedding Platonist assumptions in your question? –  Niel de Beaudrap Dec 2 '12 at 11:05
    
Sure, I'm playing the devils advocate rhetorically here. –  Mozibur Ullah Dec 2 '12 at 13:11
    
The satanic advocacy would be more apparent if I recognised any of the arguments as ones which are plausible (or even common) statements to make about mathematics. –  Niel de Beaudrap Dec 2 '12 at 13:16
    
I've discussed them on other threads here. I would agree with you that they are uncommon statements to make about mathematics/physics. I've actually suggested on another thread that mathematicians/physicists know more about philosophy, so as to understand the limitations and historical antecedents of their science. –  Mozibur Ullah Dec 2 '12 at 13:28
    
-1: This is not a discussion forum and we don't have threads here. We have questions, answers and comments, and maybe comment threads if you will. However, comment threads are discouraged, and it is suggested to use chat instead. –  Thomas Klimpel Dec 8 '12 at 9:27
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6 Answers

up vote 8 down vote accepted

Should a lawmaker or a relationship counsellor know any philosophy? After all, should they learn how arbitrary and groundless much of their disciplines are, they may be disillusioned, and thenceforth will always have a vague awareness of the fact that what they're doing is false.

The premise here is the same as your premise for mathematicians and scientists: that to be sheltered and have conviction in an uncritical tradition of thought would make them more efficient at their craft. This may be true, for what Kuhn would call 'normal' scientists, who in any case make up the majority of working researchers; and assuming that not all scientists could be 'revolutionary' scientists even if they all tried, perhaps then not everyone benefits from having the insight to be detached in the way that allowed Newton to simply posit action-at-a-distance without worrying about a mechanism, or engaged in the way that allowed Einstein to really plumb the notion that inertial mass and gravitational mass being the same. Let us try to entertain an 'average' scientist who we suppose is constitutionally unable to revolutionise science.

What would it even mean, to be unable to revolutionise science, and (for that reason) could not benefit from philosophy? Why would such an impairment exist in someone who is otherwise a good researcher, either as a theorist or an experimentalist? I find it difficult to imagine, failing some very definite theory of psychology which would allow me to see a mechanism for it to happen. So perhaps this thought experiment is not particularly useful. Indeed, revolutionising science is not a usual thing to expect from a scientist, so asking what causes a scientist to fail to revolutionise science feels as though the question is backwards. So does this mean that the notion of a "constitutionally normal scientist" is artificial? I would say, in fact, that it does.

But never mind scientific revolution — whatever it involves, it is not easy; or if it were easy, it would become banal, and we wouldn't call it revolution any more than inventing a device is revolutionary (ie. we would mainly do so as an exaggeration, for funding or marketing purposes). Let us suppose that doing "normal" science is what we should expect a scientist to end up doing in practise, and ask how philosophy could possibly be of use; and if we can't think of any, perhaps we should shelter scientists and mathematicians from any contact with it, lest it shatter their "faith" of how to operate and teach others, wilco which makes them more efficient at their jobs.

But this notion is based on a false premise — that there is a sharp dividing line between "normal" science and "revolutionary" science. Certainly, that which we call scientific revolution is uncommon; but as I've hinted above, that's because we reserve the word 'revolution' for changes of a certain magnitude which is large enough to be rare (while noting that it seems that it is easier to make small advances than large). There are advances which, while not revolutionary, were quite important, such as Maxwell formulating his equations of electromagnetism and Poincaré's early work on Topology, which required imagination, insight, and some specific principles on which to operate to extend the existing theory; and important but still lesser advances than this, such as Rutherford's proposal of atomic nuclei from surprising particle scattering results and Fourier's theory of decomposing continuous functions into combinations of trigonometric functions. There is a spectrum, all the way down to routine calculational exercises, in which imagination is required to arrive at a meaningful answer, and sometimes even to ask a meaningful question. Those without such imagination don't write papers worth publishing. And how should one direct one's imagination? One must impose certain constraints beyond those just of the existing discipline in order to exercise their imagination most efficiently. How does one choose such constraints most effectively, if not by considerations that would best be described as philosophical? Maybe knowing Popper's work (let alone Sartre or Heidegger) is not directly constructive for eg. a string theorist or an ecologist, but having a broad perspective of the range of ways to think — and forming personal judgments about them — extend the range of the imagination and helps to define it, like stretching and giving tone to a muscle.

If we're convinced some philosophy is necessary to guide the imaginative process required for research, perhaps it is not necessary to be an effective teacher, who can restrain themselves to presenting the existing discipline. But perhaps that's also already a false assumption: if your students are to be successful, shouldn't you teach intuitions as well as the discipline, so that when they (or their students) proceed to research, they already have some guide for their imagination? And of course, if they are engaged students, they will challenge the things which you teach. A teacher who shocks their students by simply pulling a formula from thin air, without a justification or intuition to help them to understand it, is a poor one. They should at least try to have enough perspective to understand that their discipline makes demands of the imagination which must be supported, or else the only students they will have are ones which accept things by rote and which cannot actually engage with it except to observe simple corrolaries to the existing results of the field.

So — some imagination and some perspective are required to be either a good researcher or a good teacher, and philosophy is one way in which these can be exercised. Is it the best way to do so? Certainly some personal, independent reflection is necessary, and it is difficult to separate foundational or exploratory contemplation of science from metaphysics, at the least. If they should decide to read any published philosophers on the matter, this may or may not be a good use of time, depending on what problems they are concerning themselves with. But thinking seriously about the edges of their field, and reading others who can communicate well about how to think about things in a very general manner, will tend to make them better at formulating their own goals, and communicating what might otherwise be very obscure and specialised results to a broader audience, which is productive to one's career as a scientific researcher.

All of the above, I present without any experimental support, but as an attempt to provide some sort of picture of what the scientific enterprise looks like to me — on the largest historical scales, and on the smallest personal scale; and it looks to me to be an enterprise which is fuelled by leaps of the imagination, both at the research and the pedagogical level, which are guided by metaphysical and epistemological principles to the extent that these imaginative leaps are not achieved by sheer luck. If philosophy is capable of disillusioning a teacher or researcher, that would suggest ipso facto that the basis for their teaching or research was an illusion, and that if they were successful, it was either through sheer luck (which by definition is unreliable) or because the principles which they used were not badly wrong, in which case further contemplation may allow them not only to be reassured, but possibly yet more effective.

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Most scientists are familiar with the scientific method; and many important, distinguished scientists can produce important results based on the scientific method. A revolution happens when a foundation of science, like the scientific method, is revised; in a revolution, not only a result is discovered, but all existing results have to be reappropriated under the new framework. Note that even though science has existed since ancient time, scientific method as we know it is relatively recent (19th century). –  Lie Ryan Dec 16 '12 at 10:40
    
@LieRyan: but even if "many" scientists produce results using the scientific method, what does this have to do with philosophical background, as distinct from an ethical recipe which has been professionally inculcated? –  Niel de Beaudrap Dec 16 '12 at 16:06
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A basic understanding of Philosophy of Science or "Wissenschaftstheorie" ("Theory of Science" which I like more) is in my opinion very important for a scientist to have an idea about:

  • What is truth? (Critique on empirical methods)
  • What can I proof and what is a proof?
  • What goals (e.g. Nuclear fission) and methods (Animal testing) of research are justifiable? (Ethics)

A very interesting questions for Mathematicians is definitely: Am I discovering or creating?

One author I really liked was Feyerabend. Also, I get the notion knowing some Philosophy keeps scientist on the ground.

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Although it's not necessarily a direct answer, I have a suspicion that the folks at this thread are discussing the issue in depth at this very moment. I believe that their insights may be of use to you.

Good luck, sorry I don't have a more direct answer for you.

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Please consider adding some excerpt from the linked article. If the link dies, your answer becomes invalid. –  bytebuster Dec 3 '12 at 5:50
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@bytebuster: if the link dies, then this question most likely also dies along with it. –  Lie Ryan Dec 16 '12 at 10:41
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Yes as you might find that some questions do not need to be answered.

It's a focussing thing. Either you see a small part and forget the effects on the whole or you see the whole but loose sight of the small parts since there are so many. Funnily enough the very old philosophy and the very new philosophy kind of agree that everything has an effect. My conclusion to paraphrase is if you can't beat'em - join'em. The question howeve is who is "'em"

Wikipedia Dao De Ching

Wikipedia Post Structuralism

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Yes.

This part of your question

Eugenia Cheng, a professional mathematician has said, that when she talks to philosophers, she's convinced that it doesn't exist, that it is problematic; but when she returns to her work, there they are, the concepts she handles, as solid as they can be before her minds eye.

Can professional mathematicians/physicists actually assimilate how problematic mathematics/physics is, if they are to do their work, if they are too see the concepts they handle with concrete clarity, rather than as misty illusions. Rather than liberating, it may be disillusioning. Like piercing the veil of maya.

suggests that the crux of the issue here is whether mathematics is analytical or synthetic,

“Analytic” sentences, such as “Ophthalmologists are doctors,” are those whose truth seems to be knowable by knowing the meanings of the constituent words alone, unlike the more usual “synthetic” ones, such as “Ophthalmologists are ill-humored,” whose truth is knowable by both knowing the meaning of the words and something about the world.

or possibly whether mathematics is formulated as an a priori or a posteriori discipline.

A given proposition is knowable a priori if it can be known independent of any experience other than the experience of learning the language in which the proposition is expressed, whereas a proposition that is knowable a posteriori is known on the basis of experience. For example, the proposition that all bachelors are unmarried is a priori, and the proposition that it is raining outside now is a posteriori.

Does this matter from a purely calculational standpoint? Not really. Mathematics functions most of the time just fine, regardless of what justifies it. However...

The philosophy of math presents some problems, but there should be no reason why it prevents normal functioning in mathematics or physics. Philosophical perspectives can even open up new ideas. For example, finitism may be able to provide a justification for renormalization, and someone even attempted formulating quantum mechanics with finitism in mind. The philosophy of science has its own questions about realism, but the questions raised by about identity in quantum mechanics could conceivably help resolve the quantum interpretation debate.

People actively working in these fields should not just know about the philosophy. They should be participating in the debates.

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It is my opinion that math does actually exist. For example in the OP I read that "3 is a prime" but 3 does not exist.

3 Does exist !

If you have 3 apples and you want to devide them " fair " between 2 persons without cutting the apples, then you will notice that 3 exists and it is " prime ".

A prime amount means it can only be divided "fair" in 2 ways ; the divisors of a prime p being 1,p.

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