MathMart

Karma: 64

MathMart 20 Jun 2026 14:37 UTC
2 points
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in reply to: kbear’s comment on: Computational models of first-order theories
The choice to use E was deliberate, in order to keep a distinction between the relation being built and actual set elementhood. I don’t know if it was the right choice, but that was the reasoning.

Computational models of first-order theories

MathMart16 Jun 2026 23:02 UTC

5 points

2 comments11 min readLW link

MathMart 4 Oct 2025 17:13 UTC
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in reply to: Aatu Koskensilta’s comment on: Why’s equality in logic less flexible than in category theory?
That’s also true in first-order logic with equality, since nothing except convention stops us from considering models where multiple objects are equal according to the equality predicate. The choice to exclude models which include duplicate objects is just a side-condition used to filter out inconvenient models when studying semantics. We can include such a side-condition when considering the semantics of set theory without equality, too, so it doesn’t seem fair to me to single it out as being uniquely incapable of defining equality.

(In fact, I’d argue that this also applies to second-order logic. Second-order logic can be given Henkin semantics, which have all the same idiosyncracies as first-order semantics. Using these semantics, we can get models of second-order logic with duplicate objects, just like first-order logic. I’d argue that standard second-order semantics are more or less just using a side-condition to filter out models which have missing subsets. But if we wanted to we could include similar side-conditions when considering models of first-order set theory, too, so it doesn’t seem fair to me to to say second-order logic can define equality while first-order logic can’t.)

MathMart 2 Oct 2025 21:28 UTC
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on: Why’s equality in logic less flexible than in category theory?
Since multiple answers here mention equality being undefinable in first-order logic, I want to say that that’s only true if there are an infinite number of constants/functions/predicates in the language. Since set theory can be formalized using only a single predicate, it is possible to define equality this way:

$x = y \equiv \forall z ((x \in z \leftrightarrow y \in z) \land (z \in x \leftrightarrow z \in y))$

(Where x and y can be the same variable, but z must be a different variable from them both)

By simply replacing every instance of the formula “x=y” with this definition, set theory can be formalized in first-order logic without equality.

MathMart 13 Dec 2023 0:53 UTC
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in reply to: Logan Zoellner’s comment on: A Philosophical Tautology

So what part of a mathematical universe do you find distasteful?

the idea that “2” exists as an abstract idea apart from any physical model

It’s this one.

Okay, but if actual infinities are allowed, then what defines small in the “made up of small parts”? Like, would tiny ghosts be okay because they’re “small”?

Given that you’re asking this question, I still haven’t been clear enough. I’ll try to explain it one last time. This time I’ll talk about Conway’s Game of Life and AI. The argument will carry over straightforwardly to physics and humans. (I know that Conway’s Game of life is made up of discrete cells, but I won’t be using that fact in the following argument.)

Suppose there is a Game of Life board which has an initial state which will simulate an AI. Hopefully it is inarguable that the AI’s behavior is entirely determined by the cell states and GoL rules.

Now suppose that as the game board evolves, the AI discovers Peano Arithmetic, derives “2 + 2 = 4”, and observes that this corresponds to what happens when it puts 2 apples in a bag that already contains 2 apples (there are apple-like things in the AI’s simulation). The fact that the AI derives “2 + 2 = 4″, and the fact that it observes a correspondence between this and the apples, has to be entirely determined by the rules of the Game of Life and the initial state.

In case this seems too simple and obvious so far and you’re wondering if you’re missing something, you’re probably not missing anything, this is meant to be simple and obvious.

If the AI notices how deep and intricate math is, how its many branches seem to be greatly interconnected with each other, and postulates that math is unreasonably effective. This also has to be caused entirely by the initial state and rules of the Game of Life. And if the Game of Life board is made up of sets embedded inside some model of set theory, or if it’s not embedded in anything and is just the only thing in all of existence, in either case nothing changes about the AI’s observations or actions and nothing ought to change about its predictions!

And if the existence or non-existence of something changes nothing about what it will observe, then using its existence to “explain” any of its observations is a contradiction in terms. This means that even its observation of the unreasonable effectiveness of math cannot be explained by the existence of a mathematical universe outside of the Game of Life board.

Connecting this back to what I was saying before, the “small parts” here are the cells of the Game of Life. You’ll note that it doesn’t matter if we replace the Game of Life by some other similar game where the board is a continuum. It also doesn’t even matter if the act of translating statements about the AI into statements about the board is uncomputable. All that matters is that the AI’s behavior is entirely determined by the “small parts”.

You might have noticed a loophole in this argument, in that even though the existence of math cannot change anything past the initial board state, if the board was embedded inside a model of set theory, then it would be that model which determined the initial state and rules. However, since the existence of math is compatible with every consistent set of rules and literally every initial board state, knowing this would also give no predictive power to the AI.

At best the AI could try to argue that being embedded inside a mathematical universe explains why the Game of Life rules are consistent. But then it would still be a mystery why the mathematical universe itself follows consistent rules, so in the end the AI would be left with just as many questions as it started with.

MathMart 12 Dec 2023 11:24 UTC
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in reply to: Logan Zoellner’s comment on: A Philosophical Tautology
My view is compatible with the existence of actual infinities within the physical universe. One potential source of infinity is, as you say, the possibility of infinite subdivision of spacetime. Another is the possibility that spacetime is unboundedly large. I don’t have strong opinions one way or another on if these possibilities are true or not.

MathMart 11 Dec 2023 5:16 UTC
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in reply to: Logan Zoellner’s comment on: A Philosophical Tautology
The assumption is that everything is made up of small physical parts. I do not assume or believe that it’s easy to predict the large physical systems from those small physical parts. But I do assume that the behavior of the large physical systems is determined solely from their smaller parts.

The tautology is that any explanation about large-scale behavior that invokes the existence of things other than the small physical parts must be wrong, because those other things cannot have any effect on what happens. Note that this does not mean that we need to describe everything in terms of quantum physics. But it does mean that a proper explanation must only invoke abstractions that we in principle would be able to break down into statements about physics, if we had arbitrary time and memory to work out the reduction. (Now I’ve used the word reduction again, because I can’t think of a better word, but hopefully what I mean is clear.)

This rules out many common beliefs, including the platonic existence of math separately from physics, since the platonic existence of math cannot have any effect on why math works in the physical world. It does not rule out using math, since every known instance of math, being encoded in human brains / computers, must in principle be convertible into a statement about the physical world.

MathMart 10 Dec 2023 16:15 UTC
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in reply to: Vladimir_Nesov’s comment on: A Philosophical Tautology
I completely agree that reasoning about worlds that do not exist reaches meaningful conclusions, though my view classifies that as a physical fact (since we produce a description of that nonexistent world inside our brains, and this description is itself physical).

it becomes apparent that if our physical world wasn’t real in a similar sense, literally nothing about anything would change as a result.

It seems to me like if every possible world is equally not real, then expecting a pink elephant to appear next to me after I submit this post seems just as justified as any other expectation, because there are possible worlds where it happens, and ones where it doesn’t. But I have high confidence that no pink elephant will appear, and this is not because I care more about worlds where pink elephants don’t appear, but because nothing like that has ever happened before, so my priors that it will happen are low.

For this reason I don’t think I agree that nothing would change if the physical world wasn’t real in a similar sense as hypothetical ones.

MathMart 10 Dec 2023 15:58 UTC
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in reply to: Logan Zoellner’s comment on: A Philosophical Tautology
I will refer to this other comment of mine to explain this miscommunication.

MathMart 10 Dec 2023 14:27 UTC
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in reply to: Vladimir_Nesov’s comment on: A Philosophical Tautology

Reasoning being real and the thing it reasons about being real are different things.

I do agree with this, but I am very confused about what your position is. In your sibling comment you said this:

Possibly the fact that I perceive the argument about reality of physics as both irrelevant and incorrect (the latter being a point I didn’t bring up) caused this mistake in misperceiving something relevant to it as not relevant to anything.

The existence of physics is a premise in my reasoning, which I justify (but cannot prove) by using the observation that humanity has used this knowledge to accomplish incredible things. But you seem to base your reasoning on very different starting premises, and I don’t understand what they are, so it’s hard to get at the heart of the disagreement.

Edit: I understand that using observation of the physical world to justify that it exists is a bit circular. However, I think that premises based on things that everyone has to at least act like they believe is the weakest possible sort of premise one can have. I assume you also must at least act like the physical world is real, otherwise you would not be alive to talk to me.

MathMart 10 Dec 2023 13:47 UTC
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in reply to: Vladimir_Nesov’s comment on: A Philosophical Tautology
Okay, let’s forget the stuff about the “I”, you’re right that it’s not relevant here.

For existence in the sense that physics exists, I don’t see how it’s relevant for reasoning, but I do see how it’s relevant to decision making

Okay, I think my view actually has some interesting things to say about this. Since reasoning takes place in a physical brain, reasoning about things that don’t exist can be seen as a form of physical experiment, where your brain builds a description which has properties which we assume the thing that doesn’t exist would have if it existed. I will reuse my example from my previous post to explain what I mean by this:

To be more clear about what I mean by mathematical descriptions “sharing properties” with the thing it describes, we can take as example the real numbers again. The real numbers have a property called the least upper bound property, which says that every nonempty collection of real numbers which is bounded above has a least upper bound. In mathematics, if I assume that a variable x is assigned to a nonempty set of real numbers which is bounded above, I can assume a variable y which points to its least upper bound. That I can do this is a very useful property that my description of the reals shares with the real numbers, but not with the rational numbers or the computable real numbers.

So my view would say that reasoning is not fundamentally different from running experiments. Experiments seem to me to be in a gray area with respect to this reasoning/decision-making dichotomy, since you have to make decisions to perform experiments.

MathMart 10 Dec 2023 13:06 UTC
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in reply to: Logan Zoellner’s comment on: A Philosophical Tautology
I don’t say in this post that everything can be deduced from bottom up reasoning.

MathMart 10 Dec 2023 0:13 UTC
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in reply to: Vladimir_Nesov’s comment on: A Philosophical Tautology
The fact that I live in a physical world is just a fact that I’ve observed, it’s not a part of my values. If I lived in a different world where the evidence pointed in a different direction, I would reason about the different direction instead. And regardless of my values, if I stopped reasoning about the physical world, I would die, and this seems to me to be an important difference between the physical world and other worlds I could be thinking about.

Of course this is predicated on the concept of “I” being meaningful. But I think that this is better supported by my observations than the idea that every possible world exists and the idea that probability just represents a statement about my values.

MathMart 9 Dec 2023 20:25 UTC
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in reply to: TAG’s comment on: A Philosophical Tautology

clearly physical brains can think about non physical things.

Yes, but this is not evidence for the existence of those things.

But it’s not conclusive in every case, because the simplest adequate explanation need not be a physical explanation.

There is one notion of simplicity where it is conclusive in every case: every explanation has to include physics, and then we can just cut out the extra stuff from the explanation to get one that postulates strictly less things and has equally good predictions.

But you’re right, there are other notions of simple for which this might not hold. For example if we define simple as “shortest description of the world which contains all our observations”. Though I think this definition has its own issues, since it probably depends on the choice of language.

Still, this is the most interesting point that has been brought up so far, thank you.

Edit: I was too quick with this reply and am actually wrong that my notion of simplicity is conclusive in every case. I still think this applies in every case that we know of, however.

Edit 2: I think the only case where it is not conclusive is the case where we have some explanation of the initial conditions of the universe which we find has predictive power but which requires postulating more things.

MathMart 9 Dec 2023 20:03 UTC
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in reply to: TAG’s comment on: A Philosophical Tautology

Whatever “built on top of” means.

In ZFC, the Axiom of Infinity can be written entirely in terms of ∈, ∧, ¬, and ∀. Since all of math can be encoded in ZFC (plus large cardinal axioms as necessary), all our knowledge about infinity can be described with ∀ as our only source of infinity.

Only for the subset of maths that’s also physical. You can’t resolve the Axiom of Choice problem that way.

You can’t resolve the Axiom of Choice problem in any way. Both it and its negation are consistent.

MathMart 9 Dec 2023 19:45 UTC
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in reply to: TAG’s comment on: A Philosophical Tautology

Again: every mathematical error is a real physical even in someone’s brain, so , again, physics guarantees nothing.

I don’t get what you’re trying to show with this. If I mistakenly derive in Peano Arithmetic that 2 + 2 = 3, I will find myself shocked when I put 2 apples inside a bag that already contains 2 apples and find that there are now 4 apples in that bag. Incorrect mathematical reasoning is physically distinguishible from correct mathematical reasoning.

There are of course, lots of infinities in maths.

Everything we know about all other infinities can be built on top of just FORALL in first-order logic.

MathMart 9 Dec 2023 19:24 UTC
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in reply to: Vladimir_Nesov’s comment on: A Philosophical Tautology
Sure, I think I agree. My point is that because all known reasoning takes place in physics, we don’t need to assume that any of the other things we talk about exist in the same way that physics does.

I even go a little further than that and assert that assuming that any non-physical thing exists is a mistake. It’s a mistake because it’s impossible for us to have evidence in favor of its existence, but we do have evidence against it: that evidence is known as Occam’s Razor.

MathMart 9 Dec 2023 19:10 UTC
1 point
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in reply to: TAG’s comment on: A Philosophical Tautology

Physics doesn’t guarantee that mathematical reasoning works.

All of math can be built on top of first-order logic. In the sub-case of propositional logic, it’s easy to see entirely within physics that if I observe that “A AND B” corresponds to reality, then when I check if “A” corresponds to reality, I will also find that it does. Every such deduction in propositional logic corresponds to something you can check in the real physical world.

The only infinity in first-order logic are quantifiers, of which only one is needed: FORALL, which is basically just an infinite AND. I don’t think it’s too surprising that a logical deduction from an infinite AND will hold in every finite case that we can check, for similar reasons to why logical deductions hold for the finite case.

It is mysterious that physics is ordered in a way that this works out, but pending the answer you say exists, it’s not any more mysterious than asking why math is ordered that way.

MathMart 9 Dec 2023 18:42 UTC
1 point
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in reply to: TAG’s comment on: A Philosophical Tautology
I haven’t used the word “reduce” since you gave a definition of it in the other thread which didn’t match the precise meaning I was aiming for. The meaning I am aiming for is given in this paragraph from this post:

If we take as assumption that everything humans have observed has been made up of smaller physical parts (except possibly for the current elementary particles du jour, but that doesn’t matter for the sake of this argument) and that the macro state is entirely determined by the micro state (regardless of if it’s easy to compute for humans), there is a simple conclusion that follows logically from that.

It doesn’t matter if we have found an explanation for consciousness yet. We still know with high confidence that it has to be entirely determined by the small physical components of the brain, so we can have high confidence that any attempted explanation will be wrong if it relies on the existence of other things than the physical components.
What links here?
- MathMart's comment on A Philosophical Tautology by MathMart (10 Dec 2023 15:58 UTC; 1 point)

MathMart 9 Dec 2023 18:34 UTC
1 point
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in reply to: TAG’s comment on: A Philosophical Tautology

There are answers to that question.

If you don’t mind, I would be interested in a link to a place that gives those answers, or at least a keyword to look up to find such answers.

MathMart

Com­pu­ta­tional mod­els of first-or­der theories

Computational models of first-order theories