After years of math classes, the crazy truth is finally coming out. It is all just in our heads. No way! How can that be? There’s an interesting debate in the world of math. Are math principles the creation of humanity, or are they universal truths that humans discovered? There are compelling arguments on both sides of the debate and both sides have several different sub-levels of thought. In this article, I will discuss them both generally.
The realists maintain that mathematical principles would exist even without people. Humans discovered the principles and brought them into practical use and any intelligent human being could also discover the same principles. This argument is supported by the fact that many cultures have discovered mathematical principles independent of one another. Also, mathematical concepts, such as the Fibonacci sequence and some fractals, occur in nature which would suggest that they exist even without people. Some realists, like the Pythagoreans, believe that the world was created by numbers. The realist point of view can lead to an almost supernatural view of mathematics.
The challenge with mathematical realism is that there is no physical domain where math entities exist. We cannot draw a perfect circle or even a line. We can conceptualize these things in our mind and we can prove them in theory; however, we cannot actually manipulate math entities in the physical world. Many math concepts exist only in the context of our understanding about them and conceptualizing them.
Another view is the anti-realists. They maintain that math is the creation of humans in order to make sense of the world. They recognize that math is an amazing, complex system and that it works as modeled by science. However, some argue that scientific principles could be explained without math. One anti-realist, Hartry Field, demonstrated this by explaining Newton Mechanics without referencing numbers or functions. He explained that, in his opinion, math is fictional and is true only in the context of the story in which it is being told.
So, is it all in our heads? A fiction that was created to explain properties in our world? In reality we may never be able to settle the debate and it may not matter. Math works. That is the beauty of it. In his article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, Eugene Wigner observes that
the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena. This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language.
Perhaps the best thing to commend mathematics as being real, is that it works. Time and time again, it works. Its principles, laws and theorems, applied over and over, in different settings produce accurate results and predictions. Einstein commented in a 1921 address titled Geometry and Experience, “It is mathematics which affords the exact natural sciences a certain measure of security, to which without mathematics they could not attain.” He explored the question of how math, a product of our mind can be so applicable to the concrete world. He asked, “How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirable appropriate to the object of reality?” Einstein answers this question with the statement, “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” He looks specifically at the field of geometry and the need humans have to learn about the relationships of real things to one another. Even though the axioms of geometry are based on “free creations of the human mind”, he says, “Solid bodies are related, with respect to their possible dispositions, as are bodies on Euclidean geometry of three dimensions. Then the propositions of Euclid contain affirmations as to the relations of practically-rigid bodies.” The abstract principles, when applied to “real” world situations prove to be accurate. Einstein continues to explain how the theory of relativity rests on the concepts of Euclidian and non-Euclidian geometry. He challenges the mind to conceptualize a universe which is “finite, yet unbounded”. In the end, it is this ability to use conceptualized principles and apply them to our world that makes mathematics work. So yes, mathematics may be all in our head and it may be a huge puzzle created by humanity, but it is effective, useful, and even beautiful.
Einstein, Albert. Geometry and Experience. http://www-history.mcs.st-andrews.ac.uk/Extras/Einstein_geometry.html
Wigner, Eugene. The Unreasonable Effectiveness of Mathematics in the Natural Sciences. http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
Wikipedia. Philosophy of Mathematics. http://en.wikipedia.org/wiki/Philosophy_of_mathematics