*How Mathematicians Think*, I was inspired by the author's position on the problem: Is mathematics discovered or created? Or, in other words, are mathematical entities and theorems objectively true or subjectively true?

I think I have encountered this question in other forms before, albeit I had no idea how to tackle it. Byers' book has shown me one possible way to approach this deep, epistemological conundrum. I have studied a little bit of mathematics and physics but I used to take the relationship between the two subjects for granted. However, after being exposed to the differential equations of damped oscillations and other applications of classical physics, I became increasingly suspicious. Mathematics, in the form of newton's laws, seems to have an uncanny ability to predict the behaviour of physical systems accurately.

*It does not need to be this way*. Certainly, laws of physics must model nature accurately to deserve the name but the

*very existence*of such mathematical laws is quite inexplicable. Byers gives one possible explanation in his book. It is a combination of Nietzschean and Platonist thought.

The central question as manifested in the fields of mathematics and physics must be discussed separately as their respective objects of study are different.

Byers claims that Mathematical truths are both subjective and objective. For example, a theorem within an axiomatic system is objectively true because it can be proven with standards of logic that are universal. Also, mathematical entities such as 'infinity' or 'zero' are clearly conceptualised in our minds and we have a strong intuition about them. Therefore, these entities are objectively "real" in a Platonic world that is wholly artificial, a construct of our intellect. They are not really there in the physical world. Consider the concept of the counting numbers, such as 'one', 'two', 'three' and so forth. How do we know that there is 'one' of something when we observe one apple, for example? It is our brain which interprets the splashes of colours in the form of sensory signals from the eyes. Our eyes obviously have limited sensitivity. That aside, the way our brain interprets these signals has been honed for the purpose of survival-- identifying predators and food for example--via natural selection. Hence our sensory perceptions are not representative of reality because they arise from our sensory organs and brain, which are machines adapted to survival. Since the counting numbers (1,2,3) arise from these inaccurate sensory perceptions, these numbers do not exist in physical reality! But the do exist in the Platonic world of mathematical forms, which mathematicians are concerned with. We have such strong intuitions about the counting numbers or integers because of the way our brain is structured. Hence mathematical entities are objective in the sense that every mathematician can envision them existing in his or her mind as a concept with certain intuitive properties. Mathematical entities are not objective in the usual sense of existing physically.

A mathematical system is composed of a set of axioms that are assumed to be valid and various theorems that are logically proven to follow from the axioms. A very well-known example is Euclidean geometry. Godel's incompleteness theorem number one claims that no mathematical system that is sophisticated enough to express number theory, the properties of integers, will be complete. This means that such a system cannot prove certain true theorems that lie within its domain of study. To prove these theorems which lie beyond its reach, the system must expand itself via modifying its axioms. However, after such a modification, the system remains incomplete as there will inevitably be a new theorem that is true but not provable in the new system. Godel's result shows that every mathematical entity will have different subjective interpretations in the form of mathematical systems and each system, being incomplete, cannot encapsulate every true theorem about the entity. For example, 'infinity' is part of both calculus and Cantor's theory of cardinal numbers but neither can express its full richness.

Mathematics is both discovered and created. I will aim to explain the unreasonable effectiveness of mathematics in physics in a later post.