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LineOfBestFit
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Philosophy of Mathematics

August 31st, 2013, 7:36 pm

I'm not sure if this will be super-clear, but I'll give it a try. I'm looking for recommendations on books which combine the philosophical underpinnings of math with descriptions of math's use. I'm not necessarily looking for a history book, but if the author takes you down those roads to better explain the philosophical reasoning for why certain methods came into practice, I suppose that would be fine.

I'm looking for stuff like, how 10 fingers on human hands became base-10 mathematical ordering, how much of objective "truth" can be understood as a result of humanity's use of math - that sort of thing. Bernstein's "Against the Gods" is a bit mass-market, but I'm on the hunt for that kind of read.

Thanks in advance for your recos.
 
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Alan
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Philosophy of Mathematics

September 1st, 2013, 12:02 am

 
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LineOfBestFit
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Philosophy of Mathematics

September 1st, 2013, 1:11 am

Thanks, Alan. That was awesome, but so brief! Left me hungry for more...
 
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Alan
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Philosophy of Mathematics

September 1st, 2013, 1:40 am

Well, here are the 1190 publications that cite it. Have fun!
 
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Cuchulainn
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Philosophy of Mathematics

September 1st, 2013, 10:17 am

http://www.amazon.com/Mathematicians-Mind-Jacques-Hadamard/dp/0691029318/ref=sr_sp-atf_title_1_1?s=books&ie=UTF8&qid=1378034128&sr=1-1

http://www.datasimfinancial.com

"Science is what you know, Philosophy is what you don't know" Bertrand Russell.
 
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Stale
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Philosophy of Mathematics

September 2nd, 2013, 1:04 pm

 
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Athletico
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Philosophy of Mathematics

September 6th, 2013, 12:27 pm

 
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LineOfBestFit
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Philosophy of Mathematics

September 6th, 2013, 11:29 pm

Cuch and Stale -- thanks, these look awesome. I suspect that for some of the forum regulars who got their math education before their investments education, these may be old hat, but for a guy like me whose path went in the opposite direction, these two works will be a fresh view of some of these ideas. Much appreciated.

Athletico -- are you recommending this out of your personal library?
 
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Cuchulainn
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Philosophy of Mathematics

September 7th, 2013, 9:33 am

The works of Georg Polya are also nice to read. here

He has steps for solving maths problems. Software developers take note. Thes methods are universally applicable.

problem heuristics

That's what happens in maths; you solve it (or not), generalise it, take a simpler case, see if you transform the problem to a known case etc. etc. Really good fun if there is time.
Last edited by Cuchulainn on September 6th, 2013, 10:00 pm
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"Science is what you know, Philosophy is what you don't know" Bertrand Russell.
 
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LineOfBestFit
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Philosophy of Mathematics

September 7th, 2013, 8:07 pm

Cuch, all of your recommendations have landed in my Amazon cart. I really appreciate your time to direct me to these. Can't wait to get started.
 
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Cuchulainn
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Philosophy of Mathematics

September 8th, 2013, 9:36 am

Of course, no discussion of Philosophy of Mathematics can ignore what Henri Poincare has to say on the issue.

Quote

Poincaré sets out a hierarchical view of the sciences, as is explicitly seen in his book The Value of Science (1905b) and is implied in Science and Hypothesis (1902). In his view the special sciences presuppose physics, which presupposes geometry, which in turn presupposes arithmetic. In his works on the philosophy of science Poincaré treats topics in this hierarchical order?first arithmetic, then geometry, then physics, etc.


Last edited by Cuchulainn on September 7th, 2013, 10:00 pm
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"Science is what you know, Philosophy is what you don't know" Bertrand Russell.
 
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Athletico
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Philosophy of Mathematics

September 9th, 2013, 12:39 pm

>> Athletico -- are you recommending this out of your personal library?

Yes, picked this up off my shelf multiple times.

I think you will like it specifically based on your statement: 'I'm looking for stuff like, how 10 fingers on human hands became base-10 mathematical ordering, how much of objective "truth" can be understood as a result of humanity's use of math - that sort of thing.'

It will probably create more questions in your mind than it answers.
 
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Athletico
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Philosophy of Mathematics

September 9th, 2013, 1:50 pm

Cuch, thanks for that link on Poincare -- good stuff.

Set theory is one place to find people really doing cartwheels in the philosophy of mathematics (Kronecker, Cantor, Brouwer, Hilbert, etc http://en.wikipedia.org/wiki/Controversy_over_Cantor's_theory). Philosophy of probability might be a close second in terms of controversy.

Poincare referred to Cantor's transfinite numbers as a disease from which mathematics would eventually be cured: "There is no actual infinite; the Cantorians have forgotten this, and that is why they have fallen into contradiction."
 
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Cuchulainn
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Philosophy of Mathematics

September 9th, 2013, 3:31 pm

Quote

Originally posted by: Athletico
Cuch, thanks for that link on Poincare -- good stuff.

Set theory is one place to find people really doing cartwheels in the philosophy of mathematics (Kronecker, Cantor, Brouwer, Hilbert, etc http://en.wikipedia.org/wiki/Controversy_over_Cantor's_theory). Philosophy of probability might be a close second in terms of controversy.

Poincare referred to Cantor's transfinite numbers as a disease from which mathematics would eventually be cured: "There is no actual infinite; the Cantorians have forgotten this, and that is why they have fallen into contradiction."


You never hear about Lagrange, Euler, Fourier, Cauchy and other orthogonal polynomial guys squabbbling with each other? Wonder why.
Last edited by Cuchulainn on September 8th, 2013, 10:00 pm
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platinum
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Philosophy of Mathematics

September 9th, 2013, 7:24 pm

Here is some interesting and perhaps timely news, from a scientific angle:

Sixth Sense: Numerosity - Science

The paper in Science is entitled: "Topographic Representation of Numerosity in the Human Parietal Cortex".

Abstract:

Numerosity, the set size of a group of items, is processed by the association cortex, but certain aspects mirror the properties of primary senses. Sensory cortices contain topographic maps reflecting the structure of sensory organs.

Are the cortical representation and processing of numerosity organized topographically, even though no sensory organ has a numerical structure?

Using high-field functional magnetic resonance imaging (at a field strength of 7 teslas), we described neural populations tuned to small numerosities in the human parietal cortex.

They are organized topographically, forming a numerosity map that is robust to changes in low-level stimulus features. The cortical surface area devoted to specific numerosities decreases with increasing numerosity, and the tuning width increases with preferred numerosity.

These organizational properties extend topographic principles to the representation of higher-order abstract features in the association cortex.

**

This perspective on numerosity as an actual sense, along the lines of sight, sound, smell, taste, and touch should be of substantial interest to neuroscientists, philosophers, mathematicians, and philosophers of mathematics alike.
Last edited by platinum on September 8th, 2013, 10:00 pm
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