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I am an formalist in the sense that I think that mathematics is just manipulation of symbols. But I think that this manipulation is motivated by the phantasy of humans: mathematical objects are for me created by human mind. What form of formalist am I?

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    what do you mean by "phantasy of humans" exactly? That's a rather opaque basis. – virmaior Jul 27 '15 at 9:40
  • I mean that mathematical objects are ideas/concepts created by the mind of humans. I mean also that some reasoning is not just syntactical reasoning but intuitive reasoning (although it can be formalized afterwards). – asdf Jul 27 '15 at 9:44
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The attitude you describe is the standard attitude of a working pure mathematician if you change a bit the weights: You can only do mathematics when you know about the semantics of your concepts.

Otherwise you make automatic theorem proving like a compiler :-)

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    I doubt it: FWIK most mathematics profess to be mathematical platonists. – Drux Jul 27 '15 at 10:28
  • Which of my statements do you consider controversial from the viewpoint of a mathematical Platonist? – Jo Wehler Jul 27 '15 at 10:52
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    A mathematical platonist would not agree that "mathematical objects are ... created by human mind" (OP's statement which you judge as standard attitude of a working pure mathematician). – Drux Jul 27 '15 at 10:56
  • I see what you mean and I agree with you, of course. Now returning to your first comment: How many of these mathematical professors consider themselves inventors and how many discoverers? Only the latter are the mathematical Platonists. Or to sharpen the question a bit: How many "working" mathematicians actually care about the difference? :-) – Jo Wehler Jul 27 '15 at 11:04
  • I don't know these specific numbers. E.g. Alain Connes strongly argues for mathematical Platonism, and he does care, but who is there to judge?. (I personally rather doubt Platonist views in general.) Bottom line: OP is perhaps an intuitionists, and in that he may be right or wrong. – Drux Jul 27 '15 at 11:09
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Another answer is that you are not a formalist at all, but an intuitionist.

There is a range of positions between Platonism and formalism that we generally choose to ignore. To my mind, intuitionism is the most promising among them.

If you think many of these 'phantasies' constitute a shared factor in human experience, then they must reflect the composite intuition humans have developed to approach measurement and combination. In that case, they are not completely formal, but are based on something real.

Accepting that mathematical entities are not ideal objects, nor simply conventions that are entirely learned, but exist in the mind, and are a shared aspect of human experience makes mathematics an aspect of psychology. Its goal is to discern the shared structure of our common genetic mental inheritance and see how its contents combine.

Initially intuitionism was motivated by an approach to exploring the weakness in negation that causes Russel's paradox. The approach was to take negation itself as a psychological habit, and not a law of nature, or simply an aspect of grammar, and to look at other forms it takes in naive interpretations, in the hope of finding an improved habit that might be less audacious, but evade the defects.

Its originator was a good mathematician, but not very gifted philosopher, and expressed his intention very poorly. So very few people grasp the approach as an alternative view of mathematics, and instead see it as a precursor to constructivism, or a weird experiment in alternative logic.

  • Thank you! I do not think that mathematics is a 'languageless activity', so I am probably NO intuitionist. – asdf Jul 28 '15 at 14:13
  • Do you absolutely disbelieve all mathematicians that claim their work is done visually, and then gets translated into words for others' consumption: not just Brower himself, but Smale, Lawvere, Thom, etc? For mathematics to have no part that is language-less, all of those folks would simply have to be lying. – user9166 Jul 28 '15 at 14:52
  • Thats their opinion, my opinion is different. You havr to accept that. By the way: do you disbelieve all the formalistic mathematicians like Hilbert, P. Cohen, von Neumann, etc? – asdf Jul 28 '15 at 14:57
  • No, but they are talking about what is important, you are telling me what mathematics is. Some differences of opinion matter a lot more than others. And their interpretation does not forbid mine. By saying mathematics is languageless, we can still use language about it productively, the same way that thoug eating is languageless, we can still discuss food and cooking, nutrition and the rest. If mathematics is not languageless, then language is an integral part of it, and what Smale is doing is not mathematics until he writes it. That discounts his entire personal experience. – user9166 Jul 28 '15 at 15:07

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