For a Platonist, on the other hand, mathematics is real. What the mathematician himself is, remains an open question in both views. A theory of consciousness is needed. TGD proposes such a theory (see for instance this and this).
In TGD, physics represents mathematics rather than vice versa. The view of the standard physicist is turned upside down. Mathematics is Platonia identified as the "world of classical worlds" (WCW) consisting of space-time surfaces in H=M4× CP2. In zero energy ontology, space-time surfaces obeying holography would represent numbers and more general mathematical objects, even theorems. Theorems can be identified as space-time surfaces which obey an almost deterministic holographic time evolution. If determinism were perfect, there would be only a single theorem - not very interesting mathematics - but this is not the case, so several theorems corresponding to different time evolutions follow from the given premises. This happens naturally. In the TGD Universe, the task of the mathematician is to become aware of the proofs that Nature produces.
This solves the problem caused by the combinatorial explosion encountered in attempts to mechanize mathematical proofs. Theorems are deduced from the axioms by rules that are analogous to the dynamic equations of classical physics. One can speak of truth-preserving dynamics. Boolean logic crystallizes these traffic rules. The problem is that these traffic rules are very weak. From given premises, a huge number of theorems can be deduced in an infinite number of ways. In practice, a mathematical machine is therefore impossible.
The combinatorial explosion can be illustrated by the set-theoretic representation of Boolean algebra. The premises correspond to the set A. The logical implication A→B corresponds to the fact that B is a subset of A. The number of subsets increases exponentially as the number of elements of A increases. There are 2N elements among the subsets if A has N elements. The result is a combinatorial explosion.
How are human mathematicians able to derive any theorems at all? Could one think that physics comes to rescue and replaces the Boolean traffic rules with holographic dynamics for proving. Could the laws of Nature define not only the axioms but also proofs of the theorems as holographic time evolutions. Given premises A would lead to a finite number of consequences B, provided that the holographic dynamics is slightly nondeterministic. The quantum-platonism of TGD would realize this dream.
See the article G&oum;del, Lawvere, and TGD or the chapter with the same title.
For a summary of earlier postings see Latest progress in TGD.
For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.
3 comments:
For a Platonist of the original Academy, mathematician is anyone searching Truth and Beauty from within, through dianoetic, intuitive and imaginary methods of gnothi seauton.
The Process Platonic ontology means, according to Euclid, Proclus and others, that the Cosmic spiritual order is holonomically and holographically present in each part - mostly instinctively. Mathematician is a poet who translates intuitive prelinguistic meaning into communicative mathematical language.
In physics, after Huygens' key discoveries and Newton's Prinicipia, the applied math based on coordinate system neusis took over the scientific rigor of synthetic geometry etc. constructive pure mathematics, and all the noise of analytic geometry caused damnatio memori to the extent that only few nowadays remember how Newton himself defined force, and Einstein was totally unaware about Huygens discovery of the tautochrone curvature (cycloid, as named by Galilei) which unifies time and gravity. And the consecutive proofs by Bernoulli brothers, Newton and others that cycloid is also the brachistochrone curvature of the least action principle.
The postmodern truth nihilistic (pseudo)axiomatics of arbitrary language games based on subjective declarations goes back to Hilbert. In Elements, Proclus uses term "axiom" for what in our textual tradition are called Common Notions. Common Notions (e.g. that whole is greater than the part) are self evident truths, and that allows to derive theorems with scientific rigor of theoretical science based on the Coherence Theory of Truth.
The true foundation of mathematics is holistic and constructive. Animistic relational process ontology of Mathematics which is a living and breathing spiritual being. Attempts to describe what became called "quantum holism" in terms of a reductionistic object-oriented theory of mathematics (applied math of engineers and physicists) is hopelessly incoherent pipe dream of false metaphysical dogma, and has lead to the current dead ends as well as endless social ills.
Wolfram's Platonism of Ruliad gives the Platonic dynamic Forms coherently contemporary meaning and terminology of programs.
Any attempt to give coordinate system neusis a coherent ontological meaning is anti-scientific to the extreme. Coordinate system point-reductionsm is a Zeno paradox to the extreme, and Zeno offered his paradoxes as reductio ad absurdum proofs against numerology of neusis method, as that leads to the Parmenidean stasis which denies any and all empirical world where experiental processes can keep on happening.
You cant "preserve" truth unless you have recognized the Truth in the first place. Sorry, but that's just how it is, and it needs to be said. Unaware of Proclus' commentary to Euclid, Bohm reached intuitively the same philosophical and ontological conclusions. And so did also Bergson - who I assume was aware of Proclus.
The hard challenge is that a serious Mathematician cannot avoid the ethical problematics of scientific theology, which forfeits the solipsism of the Lonely God and his "model theories".
PS: what do post-modern and post-truth Cantorian and Hilbertian set theorists keep on saying? That "mathematics is not science", which is an obvious falsehood for mathematics as a whole. So why does theoretical physics stay committed to unscientific math, which is ontological impossibility in any empirical world?
The deep trauma of the Eurocentric experience of the post-Cartesian paradigm manifesting as extreme subjectivism and alienation.
What we can become aware of:
Starting from minimal language {<, >} symbolizing the irreducible constructive operators of directed continuous motion (cf. mathematical/geometric time), the basic mirror symmetry of what you call ZEO breaths outwards < > and inwards > <. Yes, like all life, mathematics is a breathing spirit.
Let as start from < > as the basic (syntropic) nesting frame for string concatenation of mediants, the concrete Law of Included Middle of the holistic top-down algorithm from the top of the hyperoperation tower (p-adics were just a ladder, not the Source).
< >
< <> >
< <<> <> <>> >
< <<<> <<> <<><> <> <><>> <>> <>> >
etc.
For numeration, the tally algorithm countable elements are defined by
< <> > = 1/0 0/1 1/0
giving the numerical Stern-Brocot type structure of 2-sided signable tree with L and R sub-frames 1/0 0/1 and 0/1 1/0.
This leaves string >< as the annihilation operator of DelX operations. Lets observe that creation operator <> and annihilation operator >< - both of them iconic mirror symmetries, are not just reversible Boolean inverses, but also single bit rotations of each other, and this property extends to arbitrarly length strings ...<><><>... and ...><><><..., which are the Fibonacci-paths along binary tree when L=< and R=>.
It turns out that when breathing in from the nesting frame > <, the SB coprimes a/b become contracted to a/(b-1), the denominator element .symbolizing duration of frequency fractions. Cf. the elementary math behind Lorentzian dynamics.
Unitarity is naturally included without arbitrary axiom, search "simplicity" from the Cut-the-Knot pages.
Particle physics becomes Bohmian trajectories of L/R paths along binary tree from whole to parts, and their associated Ford circle/sphere fillings and other dynamic interactions. Rotating the circles gives cycloids and trochoids of Huygens wave mechanics made perfectly computational by inherent SB-metric of the wavelets.
FQHE non-abelian anyons 5/2 and 12/5 are the Law of Included Middle between the bosonic and fermionic aspects. Deep down in their way they reflect the only inherent numerical constants that really matter - the constants 3 and 2^3 of the cycloid curvature.
2+3=5 and
3*8=24=12*2
The solenoid of the AB-effect is very simple to construct from couple rotations of the cycloid arc. The tautochrone property has the numerical anatomy of distinct positions 2/2, 3/3, 4/4 etc. corresponding with the duration 1/1. The "oblong" fractions n/(n+1) (with simplicities 1/(n/(n+1)) ) "Lorentz" contract to the tautochrone property of the ontologically coherent holistic measurement theory, which physicists decided to ignore after Huygens and Newton, by getting blinded by their statistical mechanics.
Cycloid geometry as the holistic least action principle which rolls being towards spherical and spiraling forms solves the gravity problems as temporal least action principle instead of "pull by mass", which does not mean anything as "mass" turned out to be an undefinable ghost of incoherent fabulation of dogmatic reductionistic metaphysics of the post-cartesian methodological amputation of self-awareness from Natural Philosophy.
Lot else that could be shared, maybe you read or don't read these comments, doesn't really matter. Any case you have earned the right to hear now what has been revealed to mathematical awareness, what ever you make or don't make of what is shared here.
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