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I don't have any formal knowledge in Philosophy. I am reading a book named 'Thinking About Mathematics' by S. Shapiro. In this book I have learnt about platonism and empiricism.

Well, I think we, humans needed stimulus from the physical world to get the idea of planes, triangles, squares etc. To this point I find empiricism worthy.

But I don't see how empiricism could hold about modern mathematical objects like Mandelbrot curve, Roman Surface, Banach Space etc. I don't think we see any of these in the physical world.

So how can empiricism answer the ontology of these modern mathematical objects?

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    Platonism does not assert tat the idea of triangle and so on are derived from experience, by way of "abstraction" from the empirical objects we have "encountered": this is Aristotle. Platonism asserts that we "see", grasp, be in contact with IDEAS (i.e. Forms). – Mauro ALLEGRANZA Sep 24 '16 at 8:49
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    I think empiricism is more in the epistemology line of business : how do we come to know stuff rather than "what stuff exists". I think the empiricist would say that our knowledge of complex abstract stuff like algebras etc. is built up from knowledge of more basic, sensed stuff, kinda like an axiomatic system. iow, it's not that all knowledge is sensed, but that it is based on an empirical foundation Caveat: empiricism as a philosophical doctrine was demolished decades ago by Quine, Selkars, etc. – user20153 Sep 24 '16 at 19:43
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    Sellars, not Selkars. – user20153 Sep 24 '16 at 19:44
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    Are abstractions to be rejected somehow? Our world is built on abstractions. Law, cities, business, government, art. Those things aren't laws of physics, they're abstract human agreements. Why should math be held to a mundane standard of physicality? – user4894 Dec 9 '18 at 6:54
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When we look at the world and at our thoughts, it seems very natural to assume, that every thought is somehow rooted in the concrete reality around us. But weird objects might be constructed by combining things we already know. Banach spaces for example seem very strange, but they arise naturally when we try to understand functions.

On the other hand mathematicians might also try to combine theorems and definitions to check if they themselves are valid. New structures like the Cantor Set arise, that explain the weirdness and problems of the math we use rather than describing reality.

Maybe Hume's take on ideas is helpful if you want to think about how real those things are and how to categorize them. He tried to make empiricism work and his problem of the gold mountain seems to me very similar to your problem: How can empiricism justify complex, weird objects that seem to exist only in our imagination?

  • "But weird objects might be constructed by combining things we already know." Not good enough of an explanation because these weird objects are not combinations of things we already know. We learn about them perhaps by prototyping them as things we already know (e.g. a car is a horseless carriage). But they exist independently of the objects (e.g. cars could exist in a world without either horses or carriages_. – Tomislav Ostojich Dec 9 '18 at 3:58
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I'd like to follow-up on Hume's "complex of ideas" suggestion in @don-joe 's answer, and instead suggest Carnap's idea that "An 'object' is anything about which a 'statement' can be made", from the first page of his Aufbau. Then, depending on (your definition of) 'statement', an 'object' can be pretty much anything, including abstractions like love,hate,joy,sorrow,freedom.etc.

But 'statement' is frequently taken as 'empirical statement', in which case your domain of discourse is constrained to more 'concrete objects'. So concrete is what I think the op's suggesting. But, a la Carnap, neither mathematics nor any discipline is necessarily constrained to such concrete objects.

Mathematical axioms are frequently inspired by empirical observations, but also frequently inspired by more "abstract mathematical elegance" (or whatever you want to call it). And the latter kinds of axiomatizations give rise to more abstract kinds of objects.

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By many accounts, it can't. In order to have a fully naturalist account of mathematics, you need a middle ground between these extremes.

Kant theorized that space and time are not aspects of reality, but instead are formats that our perceptions are fitted into because of the nature of human intuition. It would then be impossible for them to be empirical, because they are necessary parts of the mind that underly our ability to accumulate data. They are, in his terminology, synthetic, a priori facts.

But that does not mean they are a part of some additional reality, as suggested by Platonism. They are instead patterns that we as a species impose on our interpretations of our environment, and that we share with one another and, indirectly, with all other related species.

(Neo-)Intuitionists generalize this to the whole of mathematics and logic, making the range of structures expressed by mathematical and logical objects part of rational psychology, rather than of ontology. Mathematical facts do not describe what is true of anything or even what is truly necessary in the world, but they do dictate what it necessary for humanity to truly grasp any aspect of the outside world and give it a shared expression.

Jungian psychology spreads this notion of shared, underlying structure beyond mathematics and into other realms like the patterns that underly recurrent religious sentiments and story structures, as the genetic component of our collective unconscious experience.

Things like the notion of the repeatability of experience, or our ideals of safety and equality, and their competing notions of the coincidental or miraculous, or the notion of heroism and earned privilege then take on the same nature as mathematical idealizations. This makes mathematics less singular of an endeavor, even if none of the other related tropes are as deeply embedded and thus our reasoning about them can never be equally clear or perfect. And it emphasizes that all human abstractions require a shared underlayment, partially inborn and partially negotiated by elaboration of conventions, in order to retain their stability.

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