Is the unreasonable effectiveness of Mathematics proof of Platonic reality
In The Unreasonable Effectiveness of Mathematics in the Natural Sciences Wigner challenges us to answer whymathematics allows us to put the ‘cart before the horse’ in physical predictions and the mysterious accuracies in the physical descriptions.
Mathematicians and physicists often talk about mathematical objects as though the exist in a ‘platonic realm’ because it offers a simple and well grounded explanation for their existence. Simply put - the attributes of these objects are a necessity… for some reason.
Where they are divided is on what that reason may in fact be, with many of the implications being muddled and misguided. Because the responsibility of mathematicians and physicists is not philosophical, almost exclusively only a caricature of ‘platonism’ is agrgued for and against.
However Plato himself was anything-but clumsy enough to make benighted, unwittingly contradictory or simple declarative statements, especially about metaphysical options. In a sense the theory of forms simply cannot be understood in purely positive terms, rather its necessity is only revealed in dialogue, this makes the theory far subtler than is often presented.
Proof of what exactly?
Broadly in science the positions are a continuum, ranging from scepticism to essentially an idealist monism :
- Constructivist - Popper’s World 3
- Logical - Frege's Third Realm
- Pythagorean - Penrose World 3
- Isomorphic - Tegmarck’s Mathematical Multiverse
For mathematical and scientific platonists, it is simply self-evident that there will be commutability between number and object, given they all partake in some way of an ideal.
Those who are sceptical of this ‘essence before existence’ do offer simpler answers to our knowledge of these ideals, we have created them. However this view struggles explaining how ‘real’ things and our constructed ‘ideals’ actually commute without being considered indiscernible identicals.
At minimum the existence before essence alternatives must account for invariant and commutable abstracts that exhibit the same syntactic interchangeability as suggested for by universal ideals.
1) Constructivist - Poppers World 3
In Poppers evolutionary epistemology ‘‘knowledge is a natural product of the human animal” and he defines theories and problem situations as products of language that posses some invariant aspect consisting of thought content.
World 3 is defined by Popper in explicit distinction to the ‘‘essentialism” of Plato. For Popper his world 3 is “the world of the products of the human mind” and therefore has a totally different metaphysic for the causal relationships with thought processes and physical bodies.
Poppers likes to point out the comparatively few problems pure math or science can meaningfully answer and attempts to exhaust these with his interplay between his other worlds. This interplay is a selective process that favours truths set up by language structures, so in a sense we only ever find what we are looking for.
2) Logical - Frege's Third Realm
Rather than the experiential ‘thought content’ world 3 of Popper, Frege appeals to a strictly rational concept akin to ‘self evidence’ for mathematical truths. Frege suggests we ‘apprehend’ something distinct between the (objective) shared world and (subjective) private world.. a third realm.
“So the result seems to be: thoughts are neither things of the outer world nor ideas. — A third realm must be recognized.” (Frege, 1918, p.302)
For Frege, thoughts comprise of a function and object, and that this says something true or false about that subject-matter. These signs may possess “interchangeability of synonymous expressions” but all denote a particular reference, regardless of presentation.
Frege’s distinguishes his theory as a mathematical platonism through his ‘apprehension’ quality. For example the relationship values of geometry are entirely independent of opinion, when grasping truth in mathematical or geometric ‘self evident’ things, we are accessing some immutable fact-of-the-matter that is neither a physical quality or dependent on mental construction.
3) Pythagorean - Penrose three worlds
Mathematical Platonists hold abstract mathematical representations as something that denote ‘things’ that are indescribable in any other form. For example entities like electrons which exhibit properties unlike anything we can ever have sensory experience, yet can be described exquisitely with mathematics .
Penrose explicitly distinguishes his world 3 from Poppers world 3 (diagrammatically) with mind independence similarities to Frege. However Penrose does not just claim “God made the integers, all the rest is the work of man” he goes a step further by privileging the complete number number line, therefore complex numbers, as a metaphysical necessity.
Penrose limits his claim on mathematics by acknowledging that meaning per se is not an entirely computable process. He attempts to address this through interactions between worlds, in this sense echoing Pythagorean’s belief that the Arche(everything) was actualised by harmony of Apeiron (plurality) and Peras (unity).
4) Isomorphic - Tegmarck’s level IV Multiverse
Tegmark takes the all previous positions platonic claims a step farther. Rather than claiming entities ‘resemble number’ Tegmark states “Reality isn’t simply described by mathematics, as physicists accept, but is, in fact, mathematical”
Tegmark considers the mathematical evidence in physical observation to be the ‘truth’ on the matter and essentially ask us - If we take the correspondence between observation and a theory seriously, then why not take a theories unobserved or even unobservable predictions equally seriously.
This leads to various extremes such as multiverses such as: Unobservable Inflationary and bubble parallel ‘multiverses’, many worlds multiverses and a mathematical landscape of various constants and solutions to equations.
All these positions are highly reasonable, it is entirely possible to argue effectively for each one and this is the source of much confusion. Each position needs to be addressed in its nuance and dismissed ONLY with respect to how plausible all the other positions are as well.
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