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Tue May 28, 2013, 01:21 PM

 

New book: Philosophy makes better mathematicians

May 28, 2013 - 4:00:00 AM

Pythagorean numbers were only positive integers, and this limited toolbox worked fine until one of his students discovered that the diagonal of a square with sides being integers could not be described only using integers. The problem was that the square root of 2 was not a rational number, but an irrational number. Immediately the problem was removed by throwing the presumptuous student in the sea, where he drowned - a radical, though not long-lasting solution. There was no getting around the fact that integers and fractions were not enough to describe the world. The mathematical way to solve the problem was to expand the concept of numbers and introduce irrational numbers. Negative numbers were also needed, but they came only later.

By University of Southern Denmark
Does mathematics consist of absolute truths, and are mathematical results always indisputable? Most people would probably respond yes without thinking twice, but the answer is actually also in part no. Mathematics can also be approached from a philosophical angle - and it is important to do so. Otherwise, we cannot ask the big, important questions in life, writes University of Southern Denmark-scientist in a new book.

The author is Jessica Carter, Associate Professor at the Department of Mathematics and Computer Science at University of Southern Denmark. She teaches philosophy, and together with Tinne Hoff Kjeldsen from Roskilde University Center in Denmark she has written about the importance of learning some philosophy and history in order to become a competent and reflective mathematician.

The book is called International Handbook of Research in History, Philosophy and Science Teaching and it will be published by Springer Verlag in June 2013. The two authors have contributed with the chapter: The Role of History and Philosophy in University Mathematics Education.

Many believe that mathematics cannot be discussed. But it can - and it is important to do so. When discussing mathematics from a philosophical point of view, we stop learning equations and formulas by heart and start talking about all the different ways we can work with mathematics and the places it can take us.

http://www.rxpgnews.com/research/New-book-Philosophy-makes-better-mathematicians_638875.shtml

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Reply New book: Philosophy makes better mathematicians (Original post)
rug May 2013 OP
Tuesday Afternoon Jun 2013 #1
struggle4progress Jun 2013 #2
Tuesday Afternoon Jun 2013 #3
struggle4progress Jun 2013 #4
Fortinbras Armstrong Jul 2013 #5
Sweeney Dec 2014 #6

Response to rug (Original post)

Sat Jun 1, 2013, 05:44 AM

1. discovered that the diagonal of a square with sides being integers could not be described only using

integers.

Could you diagram that one for me ...

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Response to Tuesday Afternoon (Reply #1)

Sat Jun 1, 2013, 06:41 AM

2. It's a bit hard to get by with only pictures here, cuz this is where geometry

and whole number arithmetic went their separate ways

It's not hard to see that one geometric square can be cut apart into four pieces and reassembled as two identical smaller squares (so it's natural to say the larger square is twice one of the smaller ones): draw the two diagonals of the larger square, then cut along them; you'll get four pieces (identical right isosceles triangles); take two and glue them together along their hypotenuses, then do the same with the other two. Voila! Two identical smaller squares from a larger one from a larger one!

Now, if you are a atomist, it is natural for you to hope that the squares are actually little square arrays of tiny atoms:

oo
oo

ooo
ooo
ooo

oooo
oooo
oooo
oooo

And since one geometric square can be twice another, the number of atoms in the bigger square should be twice the number of atoms in a smaller square. So you start looking for a square number that's twice another square number, and then youhave a lot of "close but no cigar" moments:

Two of these

oo
oo

make up one of these

ooo
ooo
oo

Oops! There's an atom missing!

Two of these

ooooo
ooooo
ooooo
ooooo
ooooo

make up one of these

ooooooo
ooooooo
ooooooo
ooooooo
ooooooo
ooooooo
oooooooo

Oops! There's an atom too many!

There's actually an endless supply of near misses like that, with an atom too few or an atom too many. Here are the first few near misses:

(3 x 3) - 1 = 2 x (2 x 2)
(7 x 7) + 1 = 2 x (5 x 5)
(17 x 17) - 1 = 2 x (12 x 12)
(41 x 41) + 1 = 2 x (29 x 29)
(99 x 99) - 1 = 2 x (70 x 70)
(239 x 239) + 1 = 2 x (169 x 169)
&c &c

But surely we can find a square number that's twice another square number, can't we?

Um ... sorry! Nope!

Here's an easy way to see there's no square number that's twice another square number. Suppose there were whole numbers A > 0 and B > 0 with A x A = 2 x B x B. Then there's a smallest such A > 0: we could find it by trying

A = 1 (nope!)
A = 2 (nope!)
A = 3 (nope!)
...

and so on until we found the very first one that worked: A x A = 2 x B x B for some B > 0. Notice that B < A. Now A is either even or odd, and if A is odd then A x A is odd and equal to the even number 2 x B x B. Since no odd number is even, we see that A must be even: say A = 2 x C. Thus 2 x C x 2 x C = A x A = 2 x B x B. Dividing by 2, we get B x B = 2 x C x C. So B is another a square number that's twice another square number. But A was supposed to be the very smallest example, and yet B < A! That's ridiculous, so there's no square number that's twice another square number

Hippasus was not popular with the atomists










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Response to struggle4progress (Reply #2)

Sat Jun 1, 2013, 07:30 AM

3. does that mean that it is Hip to be Square?



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Response to Tuesday Afternoon (Reply #3)

Sat Jun 1, 2013, 04:20 PM

4. That's my hope!

And some think it may be my only hope!

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Response to rug (Original post)

Tue Jul 2, 2013, 06:56 AM

5. One of the most important mathematical works of the 20th century

is Principia Mathematica ("Mathematical Principles" by Alfred North Whitehead and Bertrand Russell -- two names which philosophy students should recognize. It was an attempt to describe a set of axioms and rules in symbolic logic from which all mathematics could be proven.

There is a famous quote on about page 400 of the first volume, following a considerable amount of symbolic logic, "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." With the additional comment, "The above proposition is occasionally useful."

Unfortunately for Whitehead and Russell, there are a number of mathematical objections which can be raised. By the rules of the Principia Mathematica, the Axiom of Infinity (that there exists at least one infinite set) and the Axiom of Choice (that the product of a collection of non-empty sets is non-empty) are conditionals. However, the Axiom of Reducibility (required by Russell's Paradox -- the question about whether the set of sets which are not a members of themselves is a member of itself) says that they are not conditionals. Kurt Goedel's Incompleteness Theorem showed that it is impossible to derive all mathematics from a finite set of axioms. (When I was taking Introduction to Metaphysics, I had an argument with the professor about Goedel's Incompleteness Theorem and the Heisenberg Uncertainty Principle, which I maintained were philosophically significant, because they demonstrated that certain questions are unanswerable and certain facts are unknowable; he disagreed. I still say he was wrong.)

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