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August 7

How natural and universal is mathematics?

Is there anything in mathematics that has been invented by humans? To illustrate my question: suppose that on some planet there is an extraterrestrial civilisation of aliens whose brains function in a way very similar to how ours do, and the two civilisations - ours and theirs - have been developing parallel with each other, but in ignorance with each other. Then for the aliens 2 + 2 will still be 4, log₅(125) will still be 3, negative numbers will likely be identical to ours, and the ratio of a circle's circumference to its diametre will still be a constant approximating to 3.14159. Is there anything in our mathematics, and especially in the basics of mathematics usually covered in school courses, that might be different for them? Of course, exceptions would be nomenclature, terminology, symbol denomination, and a possible use of a notation other than the base-10 positional notation. --Theurgist (talk) 15:59, 7 August 2011 (UTC)[reply]

Anything that is not computable can be different. Count Iblis (talk) 17:22, 7 August 2011 (UTC)[reply]

Oh, I totally disagree. Can the collection of Turing machines that halt be different? But the halting problem is not computably decidable. --Trovatore (talk) 19:56, 7 August 2011 (UTC)[reply]

Yes, but anything we can do that has rigorous meaning is still formally describable. If we compute an integral over some interval, what we do is perform a finite number of formal manipulations. The standard interpretation that we're integrating over an uncountable number of real numbers, most of which are not formally describable is just a fairy tale. That fairy tale can be changed, what matters is that we can at most play with a countable (and in practice finite) number of formal rules. Count Iblis (talk) 21:55, 7 August 2011 (UTC)[reply]

Again, I totally disagree with you. The uncountable infinite collections are in fact real. Their existence is a falsifiable hypothesis, not falsified, with explanatory power, and is the best current explanation of the observed facts. --Trovatore (talk) 22:27, 7 August 2011 (UTC)[reply]

Whether math is empirical or a priori is a question that's debated in philosophy of mathematics. If it is empirical then there could be totally different math that we somehow can't conceive of. But I don't think it's a question you're going to find a definite answer to. That said, most of our math is based on efforts to model our empirical observations of the universe. If these extraterrestrials are in the same universe, with the same physics, I would expect they might develop a similar looking system. That's purely speculation. Rckrone (talk) 17:39, 7 August 2011 (UTC)[reply]

Empirical and a priori are not necessarily in conflict. Just because something must be the same in all possible worlds (a priori) doesn't mean that your means of finding out about it can't be empirical. I think we have an article on quasi-empiricism in mathematics; I don't think Quine or Putnam would allege that mathematics could be different than it is. --Trovatore (talk) 19:27, 7 August 2011 (UTC)[reply]

Alien ways of thought may not be identified as mathematics. Bo Jacoby (talk) 18:49, 7 August 2011 (UTC).[reply]

Yeah, a naïve question on my part, indeed. I didn't expect, though, that my illustration with the aliens would be taken so literally. The question was prompted by my observation that however advanced and complex humans' mathematic gets, it always analyses the environment's actual properties. Before asking, I tried searching Wikipedia for some articles like natural mathematics and some others, but they didn't exist. If the answer to this query requires speculation, then I should take it elsewhere, because doing speculation is not the RD's job. I'm not too good at mathematics, so you can expect some more naïve questions by me here :) Thanks for the replies. --Theurgist (talk) 19:11, 7 August 2011 (UTC)[reply]

Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk. (God made the integers, all the rest is the work of man) — Leopold Kronecker

Kronecker, of course, should not be quoted approvingly here. He was quite wrong. --Trovatore (talk) 20:51, 7 August 2011 (UTC)[reply]

... so you'd give him a delta grade for that statement? Dbfirs 23:26, 7 August 2011 (UTC)[reply]

Ich stimme zu. As a Christian, I believe God made more than the Integers, but I can understand the intention of Herr Doktor Kroenecker's statement. Isaiah 28 makes it quite plain that any clever idea man has, God has in many ways showed it to him, otherwise he would know nothing. Daniel Chapter 12 speaks of later times ( perhaps now ) when men shall run to and fro and knowledge shall be increased - written about 500 or so years B.C. This part about us running perhaps a reference to our fast paced life styles. The prophet Nahum gives a for seeing of a later event from his time of an attack on Nineveh now historical to us, which can also be a dual reference in which he says in Nahum 2:4 : "The CHARIOTS shall rage in the streets, they shall justle one against another in the broad ways: they shall seem like torches, they shall run like the lightnings." I believe God's hand is in the fact the Fibonacci Numbers are naturally occurent in Nature, in things such as the orientation of pineapple seeds and leaves on plants, and the fact the earth is on a 23.5 degree tilt, any more or less could not be good - but that is my opinion. We know of the debate as to whether Newton invented calculs or Leibniz die Rechnung erfunden hat, oder die beide sie entdeckten - that is, whether they made it or just found it. Certainly we can and should think for ourselves, and anyone who genuinely believes in God should still have enough of their own volition and free thought to work out a lot of things for themselves. In reference to the mention of Pi, I have a question. How is it Pi is an irrational number, when it exists in and of itself as a ratio ? Even if as is said, it cannnot ( as yet ) be expressed as the ratio of two wholes, how do they know so, and how is it the proof of its irrationality works when by definition Pi is only what we say it is - the ratio of a circle circumference to its diameter ? Chris the Russian Christopher Lilly 02:57, 8 August 2011 (UTC)[reply]

If you'd like to see some proofs, there is an article: Proof_that_π_is_irrational. Rckrone (talk) 05:35, 8 August 2011 (UTC)[reply]

Thank You, that was very interesting. Chris the Russian Christopher Lilly 06:46, 9 August 2011 (UTC)[reply]