Closer to Truth: Is Mathematics Eternal?

Mathematical Structure and Vedanta Philosophy

If the universe can be reduced to a pure mathematical structure, then descriptions of enlightenment within the Vedanta tradition gain a new fresh perspective

I’m about three-quarters of the way through MIT physicist Max Tegmark’s book called: Our Mathematical Universe. In it, he argues that the universe can be fundamentally thought of as a pure mathematical structure. He defines this as: a set of abstract entities with relations between them. He begins his argument by outlining three levels of reality — Internal, Consensus and External.

Internal Reality consists of our subjective perceptions of the world. It is plagued by our own personal biases, as well as the multitude of heuristics embedded in our cognition.

Consensus Reality is the world as we experience it collectively. This is reality independent of our subjective lens. It is governed by the laws of classical physics. This reality isn’t free from illusion — but it aims at being free from personal distortion. It is determined collaboratively, answering the question: what is a stable conception of the world that the majority of humans can agree on.

External Reality is the way reality actually is. Once we try to see beyond our subjective position and see through our heuristics, we can then wonder about the fundamental nature of the world. What is the basis of consensus reality?

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Tegmark claims that the link between Internal and Consensus reality is the domain of cognitive science, whereas the link from Consensus to External is the realm of Physics. His claim is that external reality is isomorphic to a mathematical structure. Namely, the features of our consensus reality, at the deepest level, are purely mathematical quantities. As an example, subatomic particles cannot meaningfully be thought to be small balls or planet-like structures. Instead, a total description of them simply involves fixing parameters such as angular momentum, principle quantum number and so on. At the deepest level, these particles are simply a set of numbers.

Mathematical Structure and Vedanta Philosophy

...Tegmark claims that the link between Internal and Consensus reality is the domain of cognitive science, whereas the link from Consensus to External is the realm of Physics. His claim is that external reality is isomorphic to a mathematical structure. Namely, the features of our consensus reality, at the deepest level, are purely mathematical quantities. As an example, subatomic particles cannot meaningfully be thought to be small balls or planet-like structures. Instead, a total description of them simply involves fixing parameters such as angular momentum, principle quantum number and so on. At the deepest level, these particles are simply a set of numbers.

I don't see that.

Various definitions of "parameter":

- a numerical or other measurable factor forming one of a set that defines a system or sets the conditions of its operation.
"the transmission will not let you downshift unless your speed is within the lower gear's parameters"

- a quantity whose value is selected for the particular circumstances and in relation to which other variable quantities may be expressed.

- a numerical characteristic of a population, as distinct from a statistic of a sample.

The existence of an immaterial parameter seems to presuppose that it (the parameter) has a separate object, system, some sort of "thing", it is a parameter of. The existential totality here consists of both that "thing" whatever it is, and its parameter(s).

I don't see that.

Various definitions of "parameter":

- a numerical or other measurable factor forming one of a set that defines a system or sets the conditions of its operation.
"the transmission will not let you downshift unless your speed is within the lower gear's parameters"

- a quantity whose value is selected for the particular circumstances and in relation to which other variable quantities may be expressed.

- a numerical characteristic of a population, as distinct from a statistic of a sample.

The existence of an immaterial parameter seems to presuppose that it (the parameter) has a separate object, system, some sort of "thing", it is a parameter of. The existential totality here consists of both that "thing" whatever it is, and its parameter(s).

Yeah I'd agree. I do think it is amusing because Materialism via Physicalism posits everything can be described by physics' mathematical net. So it naturally leads to claims like Tegmark's, that everything *is* number....but this in turn only leads to certain variations of Idealism as it's easy to grasp Platonic Math as mental objects in the mind of God...exactly as some East/West/Middle-East medievals and ancients did.

'Thus the thought [e.g.] which we expressed in the Pythagorean theorem is timelessly true, true independently of whether anyone takes it to be true. It needs no bearer. It is not true for the first time when it is discovered, but is like a [newly discovered] planet.'


– Gottlob Frege
More on Frege's arguments for mathematical objects existing in a Platonic sense can be found in the SEP.

We now describe a template of an argument for the existence of mathematical objects. Since the first philosopher who developed an argument of this general form was Frege, it will be referred to as the Fregean argument. But the template is general and abstracts away from most specific aspects of Frege’s own defense of the existence of mathematical objects, such as his view that arithmetic is reducible to logic. Fregean logicism is just one way in which this template can be developed; some other ways will be mentioned below.

'Thus the thought [e.g.] which we expressed in the Pythagorean theorem is timelessly true, true independently of whether anyone takes it to be true. It needs no bearer. It is not true for the first time when it is discovered, but is like a [newly discovered] planet.'

– Gottlob Frege

Prelude to Mathematical Neo-Platonism 

Peter Sas

This post argues for a Mathematical Neo-Platonism (MNP), where a transcendent source – analogous to the One in historical Neo-Platonism (NP) – is seen as generating the Platonic reality of mathematics, which in turn generates the physical universe in which we find ourselves. First I will discuss some interesting parallels between NP and Zermelo-Fraenkel set theory (including the axiom of Choice, together abbreviated as "ZFC"). Subsequently I will discuss the consequences of Mathematical Monism (MM) in light of the reduction of mathematics to ZFC. MM is the claim that all of reality – including physical reality – consists entirely of mathematical structures. According to physicists like Max Tegmark, MM follows from the success of modern mathematical physics, since the latter describes physical reality entirely in mathematical terms. I will argue that MM leads to MNP when we take into account the reduction of math to ZFC, where the existence of the empty set, designated by "∅", and a small number of other axioms are sufficient to generate the entire universe of pure sets in which the bulk of mathematics fits. Thus, given MM and the reduction of math to ZFC, Leibniz's famous question "Why is there something rather than nothing?" reduces to: Why does ∅ exist? And why do the axioms of ZFC hold? As I will argue in my next post, it is only from a Neo-Platonic perspective that these questions become fully answerable.

Although I agree with the argument that the success of modern physics shows that MM is true, I also think it is crucial to note that this argument limits the truth of MM to physical reality, i.e. reality as described by physics. Thus there might be non-physical realities that escape mathematical treatment. Indeed, I think this follows from the famous Hard Problem of Consciousness, which shows that the reduction of reality to mathematics stops short of how we experience reality, insofar as the qualia of that experience refuse direct reduction to non-conscious building blocks, be it the physical structures that form the human brain or the mathematical structures that model the functional organization of the brain (see Chalmers 1996). So here, in the Hard Problem of Consciousness, MM reaches its limit. However, as I will argue in my next post, this fact – that consciousness falls outside of mathematics – is precisely what will make a Neo-Platonic approach to mathematics possible. For now, however, I will abstract from the Hard Problem and simply assume that MM is true tout court. Why? Because this puts in very sharp relief the most fundamental question of ontology...

Prelude to Mathematical Neo-Platonism 

Peter Sas

Plotinus's Metaphysics of Creative (Self-)Contemplation 

In my previous post I introduced the idea of Mathematical Neo-Platonism (MNP), where a transcendent source -- analogous to the One/Good in historical Neo-Platonism -- is seen as generating the Platonic reality of mathematics which in turn generates the physical universe in which we find ourselves. This MNP is a long-term project I am working on, and I hope to write more about it in the future. But since MNP obviously harks back to historical Neo-Platonism, especially the system of Plotinus, I want to say more about the latter in this post, where I give some historical background to my proposal for MNP. I also want to discuss Plotinus because his system is highly interesting in itself and deserves much wider recognition as being the true birthplace of Absolute Idealism in Western philosophy.

Feser: Frege on What Mathematics Isn’t

Mathematics is an iceberg on which the Titanic of modern empiricism founders.  It is good now and then to remind ourselves why, and Gottlob Frege’s famous critique of John Stuart Mill in The Foundations of Arithmetic is a useful starting point.  Whether Frege is entirely fair to Mill is a matter of debate.  Still, the fallacies he attributes to Mill are often committed by others...

As Frege says, Mill’s error is to suppose that arithmetical claims are inductive generalizations from particular cases, and to confuse what are in fact applications of arithmetical truths with empirical evidence for those truths.  When we stick one pair of apples next to another to yield four apples, we are not assembling one further bit of empirical evidence in a way that gives additional inductive support for a contingent general claim to the effect that 2 + 2 = 4.  Rather, we are applying a necessary truth, and one that is already known a priori, to a specific case.  And the same thing is true of our application of the proposition that 1 = 1 to the comparison of two weights and the like.

Again, it would be a mistake to accuse Frege of begging the question against Mill.  He isn’t stomping his foot and refusing to listen to empirical evidence against a contingent generalization to the effect that 1 = 1.  Indeed, it would beg the question against Frege to characterize the situation that way, because his point is precisely that the proposition that 1 = 1 is not a contingent empirical generalization in the first place.  His point is that when Mill characterizes arithmetical statements that way, he is changing the subject.  He is no longer talking about the proposition usually expressed by “1 = 1,” but rather about some empiricist-friendly ersatz.  Mill is really just ignoring the arithmetical truth that 1 = 1 and talking instead about a very different sort of claim while using the same symbols.

Impossible Cookware and Other Triumphs of the Penrose Tile

Patchen Barss

Given that Fibonacci seems to appear everywhere in nature—from pineapples to rabbit populations—it was all the more odd that the ratio was fundamental to a tiling system that appeared to have nothing to do with the physical world. Penrose had created a mathematical novelty, something intriguing precisely because it didn’t seem to work the way nature does. It was as if he wrote a work of fiction about a new animal species, only to have a zoologist discover that very species living on Earth. In fact, Penrose tiles bridged the golden ratio, the math we invent, and the math in the world around us.

It was as though Penrose’s fanciful mathematics had forced itself into the natural world. “For 80 years, a crystal was defined as ‘ordered and periodic,’ because all crystals studied from 1912 on were periodic,” Shechtman says. “It wasn’t until 1992 that the International Union of Crystallography established a committee to redefine ‘crystal.’ That new definition is a paradigm shift for crystallography.”

Nobody knows how the story of forbidden symmetry ends. Mathematicians continue to explore the properties of Penrose tiles. Quasicrystals remain the subject of both basic and applied research. But it has been an incredible journey so far. In the past 40 years, five-axis symmetry has gone from impractical to valuable, from unnatural to perfectly natural, from forbidden to mainstream. It’s a transformation for which we can thank two scientists who pushed past conventional wisdom to uncover a remarkable new form of infinite variation in nature.

Mathematics can count "objects" but to what extent can we say objects exist apart from our own neurological interpretations of reality.  If reality is really just a blur of something unperceivable then numbers have no significance whatsoever in themselves.  As Valmar said, we invented mathematics.

Mathematics can count "objects" but to what extent can we say objects exist apart from our own neurological interpretations of reality.  If reality is really just a blur of something unperceivable then numbers have no significance whatsoever in themselves.  As Valmar said, we invented mathematics.

Even if we invent mathematics our invention could still be Eternal?

The physicist Lee Smolin uses mathematical proofs about chess as an example - did the proofs exist in some Platonic Realm before the game?

The answer to this gets deep, because we can ask what it means to be creative, whether God knows all potential paths at the least, etc.