― Philip E.B. Jourdain, The Nature of Mathematics
Have you ever considered the power of definition? As a mathematician I can define "x" to be an integer or "A" to be a set of numbers. These are mathematical objects which inhabit the sentient mind and come into existence only by definition. Now these objects may correlate to objects in reality like the mathematical object called a circle corresponds to the rough approximation to said circle that I can draw on paper. But these objects do not exist in any physical sense. There is no integer, set, or circle that I can discover in the physical world and study to determine it's properties. So are these mathematical objects simply a product of the sentient mind or do they have some kind of independent existence?
Mathematical Platonism is a Philosophy of mathematics which posits the existence of something which can be called "Mathematics" with a capital "M". This Mathematics could be described as a collection of all mathematical truths. It is an extremely large thing, this Mathematics, encompassing every possible axiom/postulate and theorem that can logically exist. Here is an example. Euclid's parallel postulate, if taken as an axiom (with some other basic axioms) generates what mathematicians call Euclidian geometry which is geometry on flat surfaces. If this postulate is modified (and some other axioms clarified) it leads to non-Euclidian geometry which is geometry on curved cur faces, like the Earth. So I can modify or invent any axiom or set of axioms and then prove theorems which are consistent with those axioms. Obviously this creates a whole universe of possibilities which, again, we will call Mathematics. Mathematical Platonism posits that this thing we call Mathematics actually exists in some way. The argument is creation versus discovery. If these objects do to exist then mathematics is simply mental creation. If the platonic mathematical world exists then mathematics is a process of discovery much like physics is the process of discovery in the physical world.
"Mathematical platonism can be defined as the conjunction of the following three theses:
Existence. There are mathematical objects.
Abstractness. Mathematical objects are abstract.
Independence. Mathematical objects are independent of intelligent agents and their language, thought, and practices." (Platonism in the Philosophy of Mathematics)
"It may be helpful if I put the case for the actual existence of the Platonic world in a different form. What I mean by this ‘existence’ is really just the objectivity of mathematical truth. Platonic existence, as I see it, refers to the existence of an objective external standard that is not dependent upon our individual opinions nor upon our particular culture. Such ‘existence’ could also refer to things other than mathematics, such as to morality or aesthetics, but I am here concerned just with mathematical objectivity, which seems to be a much clearer issue."
― Roger Penrose, The Road to Reality
In the image below, the numbers 1,2, and 3 sequentially represent the following. Only part of the platonic mathematical world corresponds to the physical world, only part of the physical world corresponds to the mental world and, only part of the mental world corresponds to the platonic world. In other words. There are a "small" amount of mathematical objects which are applicable to the physical world. There is a "small" amount of the physical world which occupies the mental world. There is only a "small" amount of the mental world which is concerned with the platonic mathematical world. This is described more completely by Sir Roger Penrose in the quote below the image.
Source for image here. Also on page 18 of The Road to Reality by Roger Penrose.
"It may be noted, with regard to the first of these mysteries—relating the Platonic mathematical world to the physical world—that I am allowing that only a small part of the world of mathematics need have relevance to the workings of the physical world. It is certainly the case that the vast preponderance of the activities of pure mathematicians today has no obvious connection with physics, nor with any other science, although we may be frequently surprised by unexpected important applications. Likewise, in relation to the second mystery, whereby mentality comes about in association with certain physical structures (most specifically healthy, wakeful human brains), I am not insisting that the majority of physical structures need induce mentality. While the brain of a cat may indeed evoke mental qualities, I am not requiring the same for a rock. Finally, for the third mystery, I regard it as self-evident that only a small fraction of our mental activity need be concerned with absolute mathematical truth! (More likely we are concerned with the multifarious irritations, pleasures, worries, excitements, and the like, that fill our daily lives.) These three facts are represented in the smallness of the base of the connection of each world with the next, the worlds being taken in a clockwise sense in the diagram. However, it is in the encompassing of each entire world within the scope of its connection with the world preceding it that I am revealing my prejudices."
― Roger Penrose, The Road to Reality
― Roger Penrose, The Road to Reality
An alternative viewpoint is put forward by Dr. Lee Smolin in the quote below. He seems to believe that mathematics, and indeed every aspect of reality, is a physical process. That is to say that mathematics is simply a construct of the mind, which is a veritable soup of chemical processes all capable of being described physically.
"Now a Platonist would say that chess always existed timelessly in an infinite space of mathematically describable games. We do not achieve anything by believing that, except an emotion of doing something elevated. Moreover, it is clear that a lot is lost; for example, we have to explain how it is that we finite beings embedded in time can gain knowledge about this timeless realm. We find it much simpler to think that at the moment the game was invented a large set of facts become objectively demonstrable, as a consequence of the invention of the game. We have no need to think of them as eternally existing truths, which are suddenly discoverable, instead we can say they are objective facts that are evoked into existence by the invention of the game of chess. Our view is that the bulk of mathematics can be treated the same way, even if the subjects of mathematics such as numbers and geometry are inspired by our most fundamental observations of nature. Mathematics is no less objective, useful or true for being evoked by and dependent on discoveries of living minds in the process of exploring the single, time-bound universe."
― Lee Smolin, Against the Timeless Multiverse
― Lee Smolin, Against the Timeless Multiverse
I think an important clue to this argument can be found in the next quote by Sir Roger Penrose.
"To say that some mathematical assertion has a Platonic existence is merely to say that it is true in an objective sense."
― Roger Penrose, The Road to Reality
― Roger Penrose, The Road to Reality
The solution to this argument seems to lie in the definition or interpretation of mathematical platonism. If it is supposed that there is literally another "world" or "universe" where mathematical objects exist then skepticism is certainly warranted. If, however, we only say that mathematical truths exist then that is another matter entirely. This is what Dr. Roger Penrose seems to adhere to. For this view to be valid logic has to hold for this universe and phenomena must be causally related in a deterministic way. In short there must be an absolute truth which is by definition not relative to the observer. It may be easy to believe that absolute truth exists for mathematical objects but many people do not believe that this applies to moral or ethical concepts. This is the domain of philosophy and theology but I would like to comment on this idea. If there are these objective truths which apply to much (if not most) of reality, physical laws and mathematical concepts for instance, then is it not logical that these truths exist for all of reality?
I think that mathematical platonism, when expressed as the existence of mathematical objective truth, is perfectly valid and philosophically sound. Now a physicalist may disagree and claim that if something has no spacial or temporal extension it does not actually exist in any sense of the word. To this person I would have to concede the following point. In lieu of physical evidence I would be eternally unable to convince said physicalist of my opinion, for even the soundest logical and philosophic argument means nothing to a them. By the definition of a physicalist there would have to be physical evidence to convince them of any proposition. If something is non-physical, such as objective mathematical truth or absolute truth, there would most likely be no explicit physical evidence. The only evidence which could be provided would be logical or philosophical. So the debate is predestined to come to a stand-still. The only path to a resolution is for both parties to accept the principles of logic and philosophy use these tools to discover an answer to the question concerning the existence of non-physical aspects of reality.
Ultimately I believe that the subjects of Philosophy, Mathematics, and Physics are intricately intertwined. If physicists can posit infinitely many universes then philosophers should be allowed to posit the existence of a soul and mathematicians able to posit the mathematical platonism. The Universe seems to be causal so therefore logic and reason should prevail in determining answers to these difficult questions. We simply have to keep an open mind and a level head.
“The Study of philosophy is not that we may know what men have thought, but what the truth of things is. ”
― St. Thomas Aquinas
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