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Mathematics: Discovered or Created?
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Math and Numbers
I have always been comfortable with mathematics but admit to not being a math wizard in school. I liked math because its rules always seemed to be consistent, predictable and dependable. Unlike English or a foreign language, math wasn't burdened with all those "exceptions to the rules". Numbers, the language of math, were never ambiguous. 32 was 32 no matter your mood, gender or where you put the accent.
As I grew older, I thought about mathematics in a different way. Specifically, I wondered if mathematics was a human invention or, like matter or energy, was something that always was. Simply put, did math always exist, being subsequently discovered by humans, or was it something that did not previously exist until we created it?
I have come to the conclusion that math is really a tool invented by people to explore the natural world. Relationships between objects and reactions to forces of nature can be explained and measured using mathematical tools invented by people. Today I don't believe math is a real eternal phenomena (like matter or energy) that was "discovered". As the great Nobel Prize physicists, Paul Dirac, once said, "..the mathematician plays a game in which he invents the rules while the physicist plays a game in which the rules are provided by nature".
Not everyone agrees with my conclusion. Some folks believe that mathematics had to have always existed. You can't "invent" 2+2=4; it just does. While you can "invent" 2+2=5, that answer would be wrong. The best response to this argument is that '2+2=4' as a result of selecting one particular set of rules to define one particular formal system.
Whatever the answer, math can be both fascinating and fun and many hours can be spent on mathematical puzzles. One of those math puzzles is Fermat's Last Theorem, a conjecture that was first proposed in 1637 and not proven until 1995. In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than two. A few years ago I was presented with what initially appeared to be a very interesting mathematical puzzle regarding numbers and factors:
To factor a number means to break it up into numbers that can be multiplied together to get the original number. Every whole number (1,2,3,etc..) has at least 2 factors; 1 and the number itself. Whole numbers with just 2 factors are called "prime numbers". 2,3,5,11,13,17 are examples of whole prime numbers. Except for the number "2" all prime numbers are odd numbers. Since all even numbers (except for the number 2) can be divided by 2, they would have more than two factors and would not qualify as a prime number. For example, the even number 10 has 4 factors (1,2,5 and 10). 1x10=10 and 2x5=10.
With the forgoing background, here was the puzzle:
Every whole number, no matter how high, has an even number of factors except for those whole numbers whose square root is a whole number. For example, all prime numbers have 2 factors (even number), while the number 10 has 4 factors and the number 32 has 6 (1,2,4,8,16 and 32). Try this with any whole number that does not have a whole number as its square root and you'll find that it has an even number of factors.
The number 36 however has a square root of 6 which is a whole number. Factors for the number 36 are 1,2,3,4,6,9,12,18 and 36. Add them up and you'll discover that the number 36 has 9 factors which is an odd number. Same with the number 64 which has a square root of 8 (a whole number). The factors of 64 are 1,2,4,8,16,32 and 64. Add them up and you'll discover that the number 64 has 7 factors which is also an odd number.
I factored all sorts of numbers in my head and on paper and found that rule holding. I thought about this strange result and tried to find a deeper metaphysical meaning. Why this strange math rule? Quite the conundrum! A couple of hours later I found the answer. I will let the reader figure it out and invite comments below.
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