-On the nature of mathematics and its claims to objectivity:
There are three primary viewpoints taken on the nature of mathematics.1
1. Platonism: Numbers and Mathematical concepts actually exist.
"Platonism is view that there exist such things as abstract objects — where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and non-mental."2
2. Nominalism: Mathematics is about objects in the world.
This can most simply be explained as the most common view of mathematics — the belief that at a fundamental level the numbers reference real objects, just like we were taught in elementary school. The number 3 therefore simply represents a collection of three apples, bottles, or whatever.
3. Fictionalism: Mathematics is fundamentally false.
This final view of mathematics is definitely bold, but it is difficult, if not impossible, to disprove. Although through this view mathematics is false, it does not dispute the fact that mathematics is useful. After all, math, which is the heart of physics, has given us much of modern technology. Fictionalism therefore draws a comparison between math and the moral principles of many religions. Just because these principles and stories may be useful and successful does not mean that they are necessarily true.
I would like to suggest that perhaps none of these views are particularly correct. Instead, a combination of these views must be considered in order to come to a conceptual understanding of the process that is mathematical thought.
In order to come to this understanding, we must first understand the relationship between math and physics. Physics is nothing more than a description of our universe. It claims no more and no less. Its purpose is to predict empirical occurrences. For example, the basic kinematic equation $v_f=v_i+a*t$ tells us that if an object is accelerating with acceleration $a$, by accounting for its initial velocity $v_i$ we can know what that object's velocity is after a certain time $t$. This final velocity $v_f$ will likely be measured in mph, kph, or m/s. And although these assignments of numbers to objects and events are of human creation, they still give us a method to describe and predict the rate of change of the position of an object without having to do an experiment every time we would like to find velocity. This is the method of discourse for all of physics. We take a specific occurrence and describe it in order to predict future outcomes. This means that for every physics equation there was a specific phenomena which we set out to model. Without the existence of this phenomena the equation would have no meaning.
Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover. -Bertrand Russell
Having a deep knowledge of mathematics is therefore essential to the physicist because it simply gives you more tools to describe phenomena accurately and efficiently. The true mystery is uncovered when we ask the question: why is mathematics so good at describing the universe?
The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning. - Eugene Wigner
Keeping that idea in mind, lets turn to nominalism. Nominalism is an easily comprehensible viewpoint on the nature of math. It references every number to a set of real or abstract objects and as such we can say that 9*5=45 because 9 sets of 5 objects gives us a total of 45 objects. Therefore, nominalism places the number line (more specifically the positive integers) as the base idea from which mathematics is drawn.
To explain where this viewpoint leads, I'm going to make up a term — the "idea set". The positive integers are an idea set because the elements of the set are completely dependent on each other. For example, the number 1 does not exist without 2 and 5 does not exist without 604. Each positive integer could not be thought of without each and every other positive integer and together they form an idea — the positive number line. This is akin to the fact that 1 degree of a circle cannot exist by itself. The degrees 0-360 are all completely necessary in order for us to create a concept of what "1 degree" is. Numbers don't exist without the number line just as degrees don't exist without circles. However, when we place the positive integers as the most basic idea set of mathematics we run into some problems. Positive integers do not necessitate negative numbers, irrational numbers, or imaginary numbers.
So lets specifically address those problems...
To the mathematical nominalist (most people), what does 6-7 mean? Can we take 7 apples from 6 apples? And when we multiply (-6)*(-5), what does it mean to have negative 6 sets of negative 5 apples? We are unable to conceptualize how a nominalistic concept of numbers can necessitate negative numbers, yet we use them regularly and are fully able to understand their worth. We can think of negative numbers as debt; a concept similar to potential energy. Just like potential energy, which represents the energy of an object or a system due to the position in space, negative numbers represent a number's position in worth. It describes the consequences of such a position. We did not invent negative numbers out of necessity; we invented this theoretical set because of its useful properties.
We can also explain imaginary numbers using the same train of thought. $i = \sqrt{-1}$. Clearly there is no physical representation of $\sqrt{-1}$. It is seemingly absurd to find the square root of a number less than zero. Well, consider the problem $x^2 = -1$ or equivalently $1*x*x = -1$. This question asks us, what transformation $x$, when applied twice, will allow you to transform 1 into -1? Here's where a spark of genius comes in. Well, what if we placed the one dimensional number line on a two dimensional plane. Can we then use this second dimension to transform 1 into -1 by using the same transformation twice? Absolutely; by simple rotation. $i$ is a transformation by rotation into a second dimension.
This second dimension, labeled the imaginary dimension, simply represents an alternate method of measuring a number(just like debt does). Therefore, when we multiply a number by $i$ we are signifying that this number is rotated 90 degrees counterclockwise using the imaginary dimension. As such, $1*i=i$. Additionally, $1*i*i$ rotates $1$ 180 degrees, transforming 1 into -1.
Positive integers don't depend on negative numbers. Similarly, positive integers don't necessitate imaginary numbers. Here, numbers are two dimensional only because it is useful. Once you create "actual" numbers through nominalistic thought, it doesn't mean that you have the basis for all of math. Not everything we use is built from that foothold. However, through the use of ideas like negative numbers and imaginary numbers, we can now mathematically represent systems which we could not have before. Imaginary numbers are just as legitimate a concept as real numbers — a tool to describe our world.
My point here is that math, just like physics, stems from goal oriented logic. Math doesn't create a singular set which is the basis for all sets so that the original set necessitates everything we can derive. It also does not claim to exist or not exist in physical or abstract form. That is not its purpose nor its rooting. Instead, math sets out to describe concepts and systems by using consistent sets that may or may not directly exist in observable phenomena. It holds the principle that logic derived from some useful tools can create as accurate a description of our world as direct observation can. Whether or not these tools are "real" or not isn't a real question. It means nothing. Math is not metaphysics. Don't defile it with such ambiguity.
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