Philosophy of Numbers

Posted by robgambrill 8 years, 6 months ago to Philosophy
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"A is A, Existence exists..."

My instinct is that numbers exist, and that they are at least similar to objects. They require consciousness to perceive them, but they do seem to exist in the world whether we notice them or not.

You might see a group of apples on the table and notice that they have the property of five. You might just ignore the number 18,156,965,975,191. Everyone else seems to, it was the amount of the national debt in dollars early this afternoon. But I have the notion that it exists anyway.

Personally, I am not sure how far I can push that argument.

Does Infinity exist? Science says we live in a finite universe, with a finite age, and a finite measure of mass in it. It is very hard to point to a concrete example of infinity.

Go down in the other direction, at certain small intervals of space of time, nothing is there and nothing happens. Can I say that the infinitely small exists?

But the concept of infinity is central to all technological progress mankind has made in the last 500 years. How can it not exist?

I know, I know, this is all rather pedantic. But the fact is philosophy and reason lie at the heart of all mathematics.

Students of philosophy often are not exposed to the fact that mathematicians are forced to look at very philosophical questions concerning the nature of the concepts they are dealing with.

Philosophy leads to logic and reason, logic and reason lead to math and science, math and science describe the natural laws governing our existence. I find it interesting that mathematics now turns back to philosophy for inspiration and understanding.

If you have never come across these ideas, I think the video is a very accessible introduction to the subject. Are numbers objects? Is math just a formal game? If non-physical A can exist, does A therefore exist?
SOURCE URL: https://youtu.be/vA2cdHLKYB8


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  • Posted by ewv 8 years, 6 months ago
    Numbers do not exist as entities in "the world", they are abstractions of human consciousness. Concepts of number are epistemological not metaphysical. You don't perceive numbers, you can only perceive some number of entities.

    The mathematical infinite is a potential, not an actual, as explained by Aristotle against Plato. That applies to both the infinitely large and infinitely small.

    Mathematics is a science of method, not a "formal game".

    If you want to understand this read Ayn Rand's *Introduction to Objectivist Epistemology". The utube speaker is thoroughly confused, and seems to realize it.
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    • Posted by $ WilliamShipley 8 years, 6 months ago
      My thought while watching was that quantities exist in the real world, but that, as you say, numbers are abstractions of human consciousness. Much like you can have horses and cows, but you don't have quadrupeds unless the human mind makes that abstraction.
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      • Posted by ewv 8 years, 6 months ago
        There are quadrapeds in reality, but reality has no 'knowledge' of that classification -- it has no knowledge at all, it just is. 'Quantities' don't exist metaphysically, only the individual entities. That there is some number of them more than one is fact, not a subjective fantasy, but identification of the particular quantity and the idea of 'quantity' are conceptual. This is an example of the principle of the 'objective' -- knowledge of the world through our particular conceptual means -- versus the 'subjective' and the 'intrinsic'. The entities we distinguish as horses and cows exist with all their attributes, however they are classified and whether or not you call all of them quadrapeds. It is a fact that each has four legs but that fact is identified and understood through our conceptual knowledge classifying the similar against the different, there is no 'fourness' or 'quadrupedness' in reality. Universals are epistemological not metaphysical.
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    • Posted by 8 years, 5 months ago
      Sorry for such a late response, I have been doing a little research on this subject, and I found a quote from Rand that made me think she wasn't too keen on Russell's paradox (see below). I didn't want to seem like I was a troll looking for a fight on the forum, so I thought maybe I should let this drop, even though I find the topic interesting.

      However, I just found a neat little article over on the Atlas Society Website. David Ross has written " Foundations Study Guide: Philosophy of Mathematics" which covers the some of the same topics in the youtube video from an objectivist perspective. He also has some more reading suggestions. It can be found here...
      http://atlassociety.org/commentary/co...


      I read "Introduction to Objectivist Epistemology" as a young person, and I saw no obvious flaws in it. Perhaps it is time to read it more critically. Ross mentions the newer 1990 edition has some comments on mathematics by Rand in the appendix. Can't wait to get the new copy and reread the book. Thanks for recommending it!

      Rand quote regarding Certainty: “Don’t be so sure—nobody can be certain of anything.” Bertrand Russell’s gibberish to the contrary notwithstanding, that pronouncement includes itself; therefore, one cannot be sure that one cannot be sure of anything. The pronouncement means that no knowledge of any kind is possible to man, i.e., that man is not conscious. Furthermore, if one tried to accept that catch phrase, one would find that its second part contradicts its first: if nobody can be certain of anything, then everybody can be certain of everything he pleases—since it cannot be refuted, and he can claim he is not certain he is certain (which is the purpose of that notion)."
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