Mathematics, despite being part of some of the most fundamental ways we understand our universe such as physics and chemistry, still remains something of a philosophical mystery. This mystery centres on numbers and their place in the world, what they are and why they function the way they do…
The second is the idea that mathematics and numbers are simply a way of interpreting and interacting with arbitrarily defined symbols, and that holds their only usefulness (36). The final is the idea of the logicians led by Russel, who claim that mathematics are useful because the universe is structured in a logically coherent way, and mathematics are thus simply an expression of that logic (36). The latter two theories have enough holes in them that they have been considered disproven, so many mathematical philosophers rely on the first, Platonic theory to describe the role of numbers in our universe. This, however, is completely un-provable, and sidesteps around the problem of describing what numbers are without actually explaining anything; if numbers are simply things that exist in another realm, that is not philosophically useful and thus not a very compelling theory. I believe that numbers and mathematics actually need to be explained by a new theory, and that while we currently do not have a completely formulated theory to explain what numbers are and how mathematics work, each of the aforementioned theories has some of the components that a complete theory of mathematics must have. The single biggest problem with the Platonic theory is its lack of utility. Its main usefulness is in the fact that it describes mathematical principles as being fundamentally true based on their existence in the Platonic realm, allowing mathematicians to pursue their goals unhindered by doubt. The theory, however, is impossible to prove, because if numbers exist in a non physical realm that has no contact with our own then it obviously cannot be observed (36). The theory that numbers exist in a non-physical realm is just as useful and un-proveable as the theory that numbers exist only in physical form inside of black holes; it could be true, but if so, who cares? This theory does, however, hold one of the fundamental principles that must be in any theory of numbers, which is the idea that numbers, even though they may certainly not exist in any kind of physical way, are real. Their interaction with the physical world, such as the fact that one can use numbers to plot a path to the moon, execute that plot and then end up on the moon, shows that there is something fundamentally real. So while the Platonic theory is fundamentally marred by its lack of usefulness and the impossibility of proving (or disproving) its accuracy, its assertion that numbers are real in some way must be part of any eventual theory of numbers. Without numbers being real things no theory of mathematics is complete. Like the platonic theory of numbers, formalism, which states that mathematics are simply a series of series of convention governing symbols (36), has both problematic and useful components for creating a useful theory of numbers. The fundamental problem with formalism is that it fails to account for the fact that, as shown above, numbers do have some relationship to reality. It is useful, however, in admitting the failings of numbers when applied to the real world. Numbers, when interacting with physical bodies, rely fundamentally on human conceptions and sensibilities. For example: when a person sees two coins, they can say that there are two coins and in some senses be correct. The problem, however, is that this relies on a human made category of what constitutes a “coin.” When someone shaves a small amount off of one of the coins, ...
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