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Old January 9th, 2017, 01:54 PM   #31
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Whether you say 9+1=10, 9+1=A, IX+I=X, θ+α=ι, S(9)=10, etc. is really just a matter of convention. The truly profound realization is, to borrow from Peano's axioms, that for any natural number n there exists a number S(n) (the successor of n), which is also a natural number and is distinct from n. And that says something fundamental about the set of the natural numbers.
I was too slow
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Old January 9th, 2017, 02:37 PM   #32
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In some circles this is a bit old hat but we are always learning I suppose and my view may well be wrong or incomplete. You could look at it as pure mathematicians prove theorems and applied mathematicians construct theories. While I would suggest one is easy to reconsile with the world we see the other is no less routed in reality. It is our understanding of reality that is insufficient to place some theorems within it or recognise their significance.
The real difference between applied and theoretical mathematicians is not in the nature of their work but in the motivation for their work. Pure mathematicians find a mathematical problem they regard as interesting and try to construct a proof or discover a counter-example. Applied mathematicians take a problem from another discipline, generally science or engineering, then they try to develop mathematical tools to solve this problem and prove that these tools are mathematically valid. At the end of the day, both are proving theorems in the same manner, it's just a question of where they came up with these theorems.

As for the, shall we call it, 'friendly competition' between applied and pure mathematicians, it's not really based on the quality of the mathematics done by one side or the other. Rather, there's a sense amongst certain theoretical mathematicians that the field is corrupted by being associated, even indirectly, with more vulgar fields, like the sciences, which admit observation as a source of knowledge rather than relying on pure logic and nothing else, as mathematicians do. It's really doesn't matter what the inspiration is for theorems, provided they can be mathematically proven to be true, the friendly competition just a way to demonstrate a philosophical mistrust of the sciences, combined, of course, with professional snobbery.

With all that said, I am a Mathematical Platonist. So I do agree that mathematics is firmly rooted in reality, in fact I think it's far more rooted in reality than our universe, which is merely a shadow of one out of an infinite number of possible realities.

Last edited by constantine; January 9th, 2017 at 02:42 PM.
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Old January 9th, 2017, 02:43 PM   #33

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If you want a proof of God, try Godel's Ontological Proof:

It relies on modal logic and the relationship between contingent truth and necessary truth. I can attest that it's a valid proof, but I will not speculate on the consistency of the axioms or the the implications. Nor will I pretend that I have fully got my head around modal logic.
I never got round that area of math.
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Old January 9th, 2017, 04:26 PM   #34

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There was a book that came out several years ago I really enjoyed It was basically like the life of Pi, about a scientist going insane trying to discover the infinite. The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity by Amir D. Aczel ? Reviews, Discussion, Bookclubs, Lists

I find interesting that so much of math is based on essential and fundamental geometric proofs. The equations in the book are a bit over my head, set theory and the concept of the infinite and such. But from what i gathered much of it is based on the simple geometric proof of a cube emerging from a single point, and that's where it sort of transcends just being math. That we can prove geometrically that 3-dimensional objects such as ourselves can emerge from a single point of existence, or something to that effect.

Last edited by Bishop; January 9th, 2017 at 04:29 PM.
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Old January 10th, 2017, 02:01 AM   #35
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Proof of god
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Well the calculator on my phone didn't come up with that.
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Old January 10th, 2017, 02:20 AM   #36

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Well the calculator on my phone didn't come up with that.
Never underestimate the powers of imaginary pies...
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Old January 10th, 2017, 05:28 PM   #37

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There are 10 types of people in this world;
Those who understand binary mathematics and those who don't....:-)
Old engineering joke.
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Old January 10th, 2017, 05:30 PM   #38

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Now seriously, I have always been awed by the computational power of logarithms
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Old January 11th, 2017, 07:10 AM   #39

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It is so awesome! What we can do with imaginary numbers and pi is totally mind blowing.
Pi is not an imaginary number - it's a ratio, and pretty fundamental to engineering and architecture.

https://en.wikipedia.org/wiki/Imaginary_number
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Old January 11th, 2017, 08:56 AM   #40

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New here, but I hope you don't mind me blabbing on my area of expertise, since this is a recent thread. (I came to history through math, oddly enough).

I'm also a big podcast guy. One thing you might like is this podcast from the BBC on the history of mathematics.

I personally wouldn't use equations as markings for important ideas in mathematics. In fact, if you consider an equation as something with symbols like plus, minus and equals in it (it is possible to think of them differently) then equations didn't exist until 1557!

I would rather use the names of individuals (I suppose the "great man/woman" approach) or publications like Euclid's elements as important markers of mathematics's influence on the world.

I could say infinitely more but I will pause here.

-Dave K
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