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I'm a maths student currently studying a philosophy of mathematics course and I'm struggling to get to grips with some terminology.

I've been told that Empiricism is the theory that knowledge is obtained a posteriori i.e. it is obtained from empirical evidence. Which from Wikipedia is said to be 'the knowledge or source of knowledge acquired by means of the senses'.

My issue is with the senses part. For example: 'scientist A says to scientist B that proposition X is true', if scientist B was an empiricist this knowledge has technically been given via a sense, would it be reasonable to allow this to be knowledge obtained empirically? Or is this relying on scientist B trusting scientist A?

This question arose in an essay I'm writing concerning the paradoxes of infinity. I.e. an empiricist would have issues with the infinite because they can't explicitly experience something infinite, a rationalist on the other hand may not have the same issues. However if a rationalist tells them the infinite is comprehensible and is well-defined, would it then be valid for empiricist to take this as truth?

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    For an empiricist, knowledge must be rooted in experience. That doesn't mean he can't trust his fellow scientist or that he has to experience everything by himself. – Olivier Mar 19 '17 at 19:30
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    See Rationalism vs Empriricsm : the Empiricism Thesis: "we have no source of knowledge other than sense experience" must be read in a broas sense. I.e. there is no knowledge that is attainable pureli a priori, i.e. by simple reflection on concepts and their meaning, whitout some sort of test against empirical facts. – Mauro ALLEGRANZA Mar 19 '17 at 19:51
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    Thus, your concern is right: is math a priori ? If not, how we can conceive the infinite from empirical sources only ? – Mauro ALLEGRANZA Mar 19 '17 at 20:07
  • empiricism as a philosophical doctrine has been discredited for decades. but i would not be so sure that we cannot "experience" the infinite. i can imagine the natural numbers; why should that not count as "experiencing the infinite"? – user20153 Mar 19 '17 at 21:23
  • I think the natural numbers are the perfect corner case of what it means to experience something. We all have an experience of the infinitude of natural numbers. You just keep adding one, and in your mind they float off into space with three dots after them. 0, 1, 2, 3, ... is a perfectly sensible notation for the intuition. This example shows that we can have thoughts, ie abstract mental experiences, about things that we have no evidence of in the physical world. So does a universally shared intuition count as an experience? Or are the natural numbers just a shared hallucination? – user4894 Mar 19 '17 at 21:50
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Your question can be answered by looking at the a posteriori/a priori distinction. We can think of the a priori realm as what can be known without relying on senses. And the really important part here is the 'can'. I didn't show that maths is after all a posteriori, because I got all my knowledge from staring at blackboards. The fact that I could in principle reach all my mathematical knowledge without ever relying on a sense is what makes math a priori.

Another way to look at it is via the distinction of genesis vs. justification. You got your mathematical beliefs from your senses, but you dont justify them with your sense-experience. When I ask you what justifies your belief that 2+2=4, you don't refer to your math teacher, but to numbers, and how they work. And that are things you've arguably never experienced.

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