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Mathematics is considered the queen of sciences as it allows us to build simplified but functional models of the reality that surrounds us.

However, I do not understand if this isomorphism could be possible if reality were not itself a consistent and axiomized formal system.

Even if we shift the focus from the mathematical representation of reality to reality itself, it is undeniable that nature follows definite rules (probably fundamental rules of which general relativity and quantum mechanics are only faces).

These rules must be defined, otherwise the reality would be inconsistent, but this does not seem to be the case. Sooner or later we should arrive at "atomic" rules (think for example of a cellular automaton) that are not derivable from other rules, but from which all the rules and the complexity of reality emerge; otherwise we would run into an infinite regression. This is what I mean by "Axioms of Reality". Are there any publications in this regard?

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    No, reality is not a theory of itself. This "isomorphism" does not extend far beyond physics, see unreasonable ineffectiveness of mathematics in biology. And that some aspect of reality is mathematizable does not lead to sweeping conclusions about its nature, it might just be an artifact of our perception and reasoning, as Kant thought. A currently popular presentation of physical universe as a mathematical structure is Tegmark's Mathematical universe
    – Conifold
    Nov 12, 2019 at 8:29
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    @Conifold Ok, but how is it possible to derive truth about reality (think for example of the discovery of antimatter by Dirac or the discovery of the Higgs bison) starting from mathematical formulas? If reality hadn't itself been a formal system, how would it be explained?
    – Yamar69
    Nov 12, 2019 at 8:35
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    Dirac's hunch was based on a theory that is now defunct (Dirac sea), as was Maxwell's derivation of his equations (ether). It was reasoning by analogy in both cases, not formal deduction. It neither started from mathematical formulas, those formulas summarized prior experiments, nor just followed them. If we manage to approximate some class of phenomena by a mathematical model we can extract some non-obvious information about them by exploring the model, but it does not mean that the model is "real". Neither ether nor Dirac sea were, and the Standard Model is also expected to be replaced.
    – Conifold
    Nov 12, 2019 at 8:50
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    @Conifold It is uncontroversially understood that the truths of thermodynamics directly follow from the principles of statistical mechanics. New concepts and structures would inevitably arise out of a set of basic principles if those principles are applied to a system with enough degrees of freedom. That means the opposite of what you suggest. It means that new and complex theoretical structures can in fact be seen as arising out of a set of simple principles (or out of one of many equivalent sets of basic principles).
    – ACat
    Nov 12, 2019 at 21:17
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    @Conifold I don't think we can derive those truths only after we explicitly name the entropy. No. All the truths about all physical experiments are already determined by statistical mechanics whether or not we name the associated quantities. I agree that we get interested in certain implications of statistical mechanics more than others and that is the reason why we name certain things and not others. But us naming things is really not the point. The point is that all those truths, in their own right, irrespective of our interest in labeling them, are determined by statistical mechanics.
    – ACat
    Nov 12, 2019 at 21:58

7 Answers 7

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A very interesting question.

I would agree that Reality may be modeled as a formal axiomatic system. There are various axioms that could be used to ground the system. One that works would state 'The Universe is a Unity'. From this follows the undecidabilty of metaphysical problems and an explanation of everything.

To make sense of this would require a lot time examining what the words in the axiom mean and how the system works, but it does work.

Interestingly, the system that derives from this axiom is not typical (for reasons too difficult to discuss here) and is not subject to incompleteness.

This is the metaphysical scheme of Middle Way Buddhism, Taoism and more generally Mysticism or 'non-dualism'. It is not often noticed that it is a formal axiomatic system.

There is far more that could be said. In Buddhism the work of Nagarjuna might be interesting to you. He explains the logic of the system. He rejects the use of the word 'Unity' but it's fine if the meaning is carefully defined. He would avoid the dangers of this word and his axiom would state 'Nothing really exists or every really happens', but this is equivalent to an axiom of Unity.

I wrote my first dissertation to prove that the 'mystical' description of Reality takes the form of a formal axiomatic system and have received no serious objections to date.

You ask about publications. I know of no published discussions on this issue. There is a vast amount of relevant literature but discussions of the exact issue are rare.

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    I would like to read your dissertation.
    – Yamar69
    Nov 12, 2019 at 14:03
  • This is fine. +1. And can be found also in the western tradition. See spinozism
    – Rushi
    Nov 12, 2019 at 14:07
  • @Yamar69 - I'd like you to read it and comment, but I should not post my own links here. If you can give me an address I'll send one.
    – user20253
    Nov 12, 2019 at 14:33
  • PeterJ: They are rare, yet it is probably one of the more basic questions philosophy has looked over.
    – Eodnhoj7
    Nov 13, 2019 at 0:26
  • @Yamar69 Hey did you read user20253's dissertation. I am really interested in this topic myself and would like to read this dissertation.
    – eeqesri
    Mar 13, 2023 at 4:32
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You can look at people like Max Tegmark, and Eugene Wigner. Perhaps you can get more information sympathetic to your point of view starting a search from those names. But I totally disagree with them.

From an intuitionistic point of view, mathematics is just psychology. Logic (and the rest of math) is what we use to describe things. Therefore, it should not be surprising that it applies to everything we can describe. We have evolved to survive in the world as it is, so our expectations and our explanatory powers have evolved to describe the world as it is reliably up to some point. We just define logic to be the stuff that lies below that point. The impulses that more often betray us get classified as physical and not mathematical facts. But there is not really an essential difference. (Intuitionism therefore counsels us to be quite careful about throwing around concepts like universality, absolute negation and infinity, and to distrust math that is too arrogant.)

Axioms are a useful way for humans to communicate about their expectations. But, in fact, no part of our knowledge, including mathematics, is actually well-founded and made up of basic principles. This notion appeals to us, but it fails. The Munchhausen Trilemma really does apply, even to math. Axioms are based on something, or they produce beautiful and useless information. So they are not basic principles. They are chosen to express useful ideas.

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  • Tegmark is well-known for crackpot theories that are obviously ridiculous to any logician. Wigner is not, but is still woefully ignorant in foundations of mathematics and hence makes erroneous claims. And you, like many other philosophers, think wrongly that there is a problem with classical FOL, when actually the problem lies in the blatant circularity in your assumptions (e.g. impredicative constructions).
    – user21820
    Jul 7, 2022 at 12:38
  • @user21820 - What do you think is "crackpot" in Tegmark? Do you think the mathematical universe idea is more absurd than David Lewis' modal realism, which is not generally treated as a crackpot idea by philosophers (even if most do not endorse it)?
    – Hypnosifl
    Jul 8, 2022 at 14:16
  • @Hypnosifl: I do not understand why you are asking me questions since you don't seem interested in learning enough logic to understand my answers. Previously, I gave you highly detailed explanations and guidance, but I do not feel that my help is appreciated.
    – user21820
    Jul 8, 2022 at 14:42
  • @user21820 I didn't recognize your username. In any case, on the other thread I asked for a layman's summary of your criticism of my post (not of your entire proof), and asked some specific yes-or-no questions to help clarify, and you refused to answer; here I'm asking for a similar basic summary of what criticisms of Tegmark one could make that wouldn't apply to Lewis. I don't think it's realistic to expect that anyone who engages in discussion with you on a philosophy forum will have time to follow a long and detailed proof on a mathematics forum, and if not they're being "unappreciative".
    – Hypnosifl
    Jul 8, 2022 at 15:14
  • That's false. The last question you asked was "does that mean the proof does not claim there is any WFF in first-order arithmetic that [Th(ℕ)] is unable to prove?" and I explicitly answered that my proof "obviously will not claim that Th(ℕ) is unable to prove some true arithmetical sentence" together with full explanation. If you didn't understand that or subsequent explanatory comments, you could have asked further (and you still can in that chat-room while it is not frozen). But you haven't.
    – user21820
    Jul 8, 2022 at 15:21
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It's interesting to note Wigner's drawing attention to the unreasonable ineffectiveness of mathematics outside of physics

I think you make a mistake, confusing axioms, and 'atomic rules'. To understand the modern use, we should look at how it evolved, from Euclidean geometry, where it's axioms were seen as 'self evident' elementary propositions. Geometry was considered the fundamental strata of mathethematics at least until Newton's time:

"Newton was convinced that only geometrical (as opposed to algebraic) proofs can be considered certain, and indeed he recast even the mathematics of Principia in geometrical garb (using the synthetic method of fluxions). Favoring geometrical techniques was part and parcel of his ideal of injecting certainty into natural philosophy; in this he saw himself in opposition to the “skeptical probabilistic” attitude of many members of the Royal Society (such as Robert Hooke and Robert Boyle)." - from a review of Guicciardini's book on Newton

Essentially Euclidean axioms are assumptions, and modern mathematics like the development of alternative geometries from the 19th century on, revealed that there are alternative sets of these.

"As used in mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (A and B) implies A), while non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic).

"When used in the latter sense, "axiom", "postulate", and "assumption" may be used interchangeably. In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., parallel postulate in Euclidean geometry). To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms), and there may be multiple ways to axiomatize a given mathematical domain. - from Wikipedia on axioms

It's notable that many historical proofs relied implicitly on commutation, but non-commutative mathematics has proven essential in quantum mechanics. I take the first type of axiom to be definitional, the second to be assumptions.

The axioms of general relativity have been given as

General relativity can be constructed from the following principles:

  • The Principle of Equivalence
  • Vanishing torsion assumption (∇XY−∇YX=[X,Y])
  • The Poisson equation (or any other equivalent Newtonian mechanics equation) -as discussed here

This has a specific aim, reducing the assumptions to a minimum and providing a basis to demonstrate consistency of deductions of the system with the axioms, without producing contradictions.

The axioms of Quantum-Field theory are still disputed.

There are issues for the axiomatising method in general posed by Godel's results, which show there are consistent sets of principles which cannot be recursively axiomatised, ie. which statements are theorems cannot be determined by an automatisable method. Godel Incompleteness puts a fundamental limit on what can achieved with axiomatising, ending this part of the objectives of the Hilbert programme and of the Principa Mathematica. Stephen Hawking clearly stated he believed Godel's results made a Theory Of Everything impossible.

The most convincing account of how mathematics gets results about the world is I think Nancy Cartwright's How The Laws Of Physics Lie. We make abstractions, and deduce results which can only be as reliable as the abstractions are valid.

I would return to geometry to understand what abstractions are. We can see how symmetries provide economy in describing things: a sphere can be described with two numbers and greatly simplifies calculating the moments of inertia and centre of mass of a body, say.

The Bekenstein Bound shows us there is a maximum amount of entropy possible in a space, and that this occurs with blackholes. This means they are the most disordered possible system, the least economy can be achieved using symmetries. This contrasts with the 'no hair' theorem, and has led to the holographic principle and suggested extension of the principle of conservation of information to a universal law (& presumably like all conservation laws, there is an associated dimensional symmetry). Look at Conformal Cyclic Cosmology, it is suggested when the universe has decayed into only photons, they don't experience time, and by geometric arguments this is equivalent to a Big Bang, or whitehole. It can be described by photon energy-density only at that point.

So, we have systems of simplified explanation, in which we seek to have minimum assumptions, and no self-contradictions. Will there be alternate systems? Clearly, like the different systems of geometry. I would suggest what is happening is a fractal process of increasing disorder between these 'absolute' information states, with emergent complexity found in systems with fractional dimensions - like the embedding of our 4D space in a 5D one in the holographic principle. I would suggest these economies of explanation/account are not fundamental, but about emerging symmetries that represent relative order or complexity within the system, such as can preserved by biological systems consuming local Gibbs Free Energy, preserving a locally ordered system which would otherwise decay into a disordered one.

Economy of axioms as I see it in this picture, is like the attempt to reduce the fundamental constants, which is to say to understand tbe point on the universe timeline when it could be described with the greatest economy. One suggestion is that we can explain many fundamental constants as a fracture plane within the E8 hyperobject, which would reduce our universe's initial conditions to a sponteneous symmetry-breaking event.

When you say

"otherwise the reality would be inconsistent"

what you really mean is, a situation would occur like the anomolous orbit of Mercury, or the Ultraviolet Catastrophe in modeling atoms - we would know our model lacked key qualities to account for inconsistencies. And we would amend the model, and reconsider the set of minimum assumptions, that we call axioms.

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SHORT ANSWER

Generally, philosophers hold the view that reality is not sufficiently described by axioms, and that self-evident axioms are difficult to prove as being somehow fundamental.


LONG ANSWER

"The map is not the territory it represents...". - Alfred Korzybski

Since we tend to conceptualize in words, it is tempting to confuse words for real things, and the relationship between words and meaning, particularly in the symbol grounding problem, is a matter of great philosophical interest, on par with various ontological questions such as Cartesian duality. Axioms, after all, are seen as 'starting points' in the process of inference and are tackled in philosophy of mathematics and the philosophy of science which both deal in natural and formal systems that incorporate axioms, which are the foundations of theories. Where the two studies differ is whether or not they address issues of natural language.

Reality, and questions about it properly conceived as metaphysics, is often understood in two ways, one as an external spacetime in which we all exist, something that is objective, and one that is a cohesive whole of what we think about what exists. These two basic perspectives are generally understood in terms of objectivity and subjectivity, although there is a third perspective called intersubjectivity that looks to reconcile the two philosophical positions.

The question of what adequately constitutes either is a very controversial metaphysical question with a range of ontological commitments and epistemic attitudes characterizing various philosophical positions. However, common to almost all of these is the idea that language, whether natural or formal, does not capture all there is to either type of reality.

For instance, the ordinary language philosopher Gilbert Ryle, railed against knowledge as being wholly linguistic, in his Concept of Mind, where he distinguished between knowledge-how and knowledge-that, where the latter is essentially the representation of reality by utterance and logical proposition, and the former is a skill which is a type of behavior. He believed strongly that there is an intellectual bias to presuppose that propositional thought is somehow interchangeable with reality in the metaphysical presuppositions of philosophers. These are largely philosophers of the analytical tradition, and there are other schools of philosophy, such as Zen Buddhism, which actually questions if reality may be described by words at all; adherents often struggle with koans to demonstrate the limitation of language.

As for the idea that somehow a certain group of propositions is somehow fundamental to reality itself, a brief survey of the history of philosophy demonstrates the controversy inherent in declaring any truth above the need for justification, and in fact, relates to the concept of the Agrippan Trilemma which suggests that justification is not wholly a process of rational discourse. This question of whether or not any metaphysical position is dominant is also a question for those who study metaphilosophy, which is the metaphysical introspection of what exactly constitutes philosophy to begin with, and how it differs from other methods such as the scientific and mathematical ones.

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Space is the universal axiom that cannot be doubted as to doubt leaves a perceivable sense of emptiness in meaning...and we cycle to the original premise again as we are left describing intuition under spatial terms.

  1. All counting is grounded in forms, where number and form are inseperable as all numbers as entities for counting equating to a form. The simplest form being the number line.

  2. All logic is grounded in forms as the variables are connected to empirical qualities or time itself which is composed of interplaying forms.

  3. All math/logic are inseperable from forms as the symbols which are attached to the forms (being it 1 orange or A=Horse) in turn follow a "formality"...this sounds like a pun but it is not. A form of reasoning is what justifies math and logic.

  4. This form of reasoning is ground in assumed points of reference that follow a linear form. One symbol progresses to another which the progression of the symbols showing how they connect. One symbol goes to many. Logic and math have an inherent underlying spatial form which lies underneath there nature simply by observing its tautological nature.

The number line, the foundation for counting and hence proof, is not only pure form as space but shows how the numbers themselves, such as 1 and 0, existing in a progressively self referencing manner where all numbers are variations of 1, with each number being a variation of further numbers prior to it as well as one.

It is spiral.

Dictionary definitions follow the same spiral pattern as well as well most logical progression.

So when dealing with axioms, we are left with form being self evident strictly because it just "is". Even the nature of the axiom, or self evidence, is assuming an assuming as knowledge and we get an abstract circularity in the nature of assuming.

Any emotion, that we do not perceive as logical, is described under spatial terms: up/down, full/empty, fragmented/unified, etc.

So as to your questions on publications? The works of Plato and his theory of forms, but generally speaking I have not heard of a philosopher dealing with the approach above except Hall briefly and Jung to some extent.

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  • There are various proposals that space is emergent, that it's a system of continuous symmetries following Noether's theorem, and in quantum gravity proposals like Loop Quantum Gravity that these arise from a more fundamental layer. There are also indications of additional dimensions at high energies. Is the imaginary number line real or not? It is essential to describing quantum fields. Form is 'self-evident'? What about the Higgs field?
    – CriglCragl
    Nov 13, 2020 at 0:14
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The axioms of reality are called "history". This forms the basis and are, necessarily, a priori, before reason because 1) reason requires language and 2) language did not exist until long after history (the "priors") began.

Quod Erat Demonstrandum

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    QED? That's not a proof. But then again this person claims a proof of god in their profile, so... Nov 21, 2019 at 22:42
  • That's why i spelled it out for you: proof by demonstration.
    – Marxos
    Nov 12, 2020 at 22:30
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It seems obvious that any system must be based on a set of strict and unmovable rules. otherwise, Nothing can guarantee the stability or even the existence of the system. (The fact that those are axioms or not is just a naming/definition problem of low interest.)

HOWEVER... those rules can be extremely low level (on the Plank scale or even lower) and unknown (if ever knowable). And we have no idea of the number of pseudo & apparently axiomatic layers which could exist between those basic rules and what appears to us as the fundamental rules

In fact, in my opinion, it might be nothing more that something that produce chaos and some kind of causality, and then all subsequent rules and layers of rules are decided by "natural selection" (which arises as an immediate consequence of the existence of this causality⁽¹⁾.
(1) note that I explicitly say causality and not time, because special relativity shows through the rejection of simultaneity, that time is most probably a much higher level concept than this causality)

This natural selection means that:

  • perpetuating thing (which are preserved through causality) are favored against unstable things (which are not preserved);
  • reproducing things are favored over non reproducing things.
  • rules that help things to survive are themselves perpetuated by their support (the things that survive thanks to them).
  • The fact that some things are perpetuated is decided by the preservation of invariant - this is more of a way to understand the previous rules than a rule by itself.

In fact, re-reading your post, I realize that What I wrote up to here is nothing more than what you stated in your question. The problem here is how to avoid the infinite recursion. I thing the solution is quite straightforward from here :

  • You cannot have arbitrary axioms, because it would only forward the problem to WHO decided those arbitrary axioms, and does this “who” have axioms ?
  • You cannot have infinite recursion with no starting point because it would make the system have no foundation (it could be infinite in the sense of ordinal sets - i.e. the causality chain between the foundation and here might be infinite, but it might not be without a foundation)
  • So the only remaining option is the one of the cycling foundation. i.e.: the very foundation of reality must be defined only by a set of relationships between fundamental things and NOT by defining those fundamental things.

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