Where do numbers actually originate from? Is mathematics just an abstract game for intellectuals or does it have some greater meaning? Could we ever physically calculate number pi? These are questions that have continually plagued mathematicians although have no determinable or absolute answer. The importance of mathematics is widely acknowledged, considering that without it, science would lack the fundamental base that provides proof and certainty when observing our environment. This allows us to illustrate patterns and trends in constant laws which aren’t relative, but are certain and consistent. Mathematics, in its rudimentary form, is an abstract series of axioms or basic assumptions that must be followed, like a chess game with a simple set of rules that must be known before one can play. Whether these axioms are inherent or whether they can be accredited to human invention, until proved or disproved, remains conjecture for many.
G. H. Hardy stated: “I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our creations, are simply the notes of our observations.” which essentially summarizes Platonism. Numbers, being abstract entities that exist in their own unique sense, are eternally static and exist to describe the natural world. An effective way of validating this assumption would be analysing the radius of a circle. It is universally recognised that to find the area of a circle we use pr2, but if there wasn’t an entity to “name” the area, would the value still exist? Would it remain unaffected by the mental or physical observation of humans? Platonism argues that numbers are the solutions to all uncertainty, that they are the language in which we interpret and understand or the natural world with pure certainty and clarity. That numbers exist in an intangible, inimitable form and can therefore unfortunately never meet logarithms.
Werner Heisenberg highlights the intrinsic link that mathematics and the science of nature really have; it shows that mathematics has the ability to predict things about the world that we do not yet know. This could arguably be rationalized by Platonism for it suggests that mathematical entities are “alive” and perpetually constant in some unique form. The links and connections between theory and reality exist within our observable reality and are waiting to be discovered.
There are, on the other hand, many criticisms of Platonism, namely the contradictory concept within mathematics itself: infinity. Following the aforementioned logic, one would be inclined to think that because there are infinitely many numerals, that there should be infinitely many unobservable entities. This seems to defeat the stated objective, for this almost entirely removes absolute value and means that we are led dangerously into mysticism.
Platonism also implies fixed, definite rules that are constantly and forever true throughout the known universe. These rules follow the five Euclidean geometric standards:
A straight line segment can be drawn joining any two points
Any straight line segment can be extended indefinitely in a straight line
Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center
All right angles are congruent
If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate
The fifth axiom in particular has been considered outdated by some. The fifth point explains that two lines on a triangle couldn’t possibly be parallel (inevitably must intersect) because the two angles add up to less than 180 degrees. Candidly, non-Euclidean geometry, in particular Riemannian (1822-66), proved that lines of a greater sum than that of two right angles could still intersect, but on a curved surface. This was, during the time of discovery, greatly ignored and suppressed as it had the potential to undermine the widely accepted platonic view, and it was unheard of that there could be uncertainty and inconsistency within mathematics.
It is strange that objects with such an ideal existence can describe so much of the physical nature compared to abstract perfection, such as perfect circles. Do these exist in reality, or are they an abstract thought with no real connection to actuality? Einstein once queried “How is it possible that mathematics a product of human thought that is independent of experience, fits so excellently the objects of physical reality?” His opinion is wholly different to that of Plato and his followers: Formalism, the opposing view, argues that mathematics is an invented set of tools, that it is a “game played according to certain simple rules with meaningless marks on paper” (David Hilbert).
I think that modern physics has definitely decided in favor of Plato. In fact the smallest units of matter are not physical objects in the ordinary sense; they are forms, ideas which can be expressed unambiguously only in mathematical language.
How big is the biggest number you can imagine? Now imagine multiplying this by 2. Can you still imagine it? David Hilbert was an early 20th century German mathematician who believed arithmetic was finite because the mind had restrained capabilities, in the sense that we can only imagine up to a certain scale and as such we are only able to put them into limited context. His “proof theory” then explores the axioms and the rules of consistency within mathematics. This ideology supports the formalistic views as it implies that we form these consistencies so that we can play the game of math. Imagine a game of chess, if the rules or consistencies and axioms were all destroyed, would the game of chess still exist out there? Formalists, such as Einstein, argue that if no one knows how to play, then the game surely cannot exist: the entities would only appear in ones mind and must be invented to exist. Yet this philosophy lacks the satisfactory explanation for the uncanny connection that this abstract “game”, as it were, relates to real life.
The most satisfying and explanatory conclusion could only arise from an indistinguishable amalgamation of the elements of these theories. We humans choose which theories and equations to study and we apply them to the things we see around us. Could it be that mathematics is an invented science with subtle links to nature (abstract entities) that must be discovered? To what extent can even the best proposals ever be free from contradiction? Furthermore, how can mathematics, considered the most certain and objective science, also fall victim to proofs by negation? Perhaps even the most experienced mathematicians and physicists will never be able to explain how these correlations have arisen, but the privilege of being able to clarify and truly understand them is as good a reason as any to keep trying.