I don’t think these are free parameters in the same sense.

Like, if one theory says that a hunk of metal actually is made of many microscopic grains of various sizes and orientations, where the sizes and orientations of these grains has an effect on the behavior of the metal, you don’t count the “the sizes and orientations of these grains” as free parameters, do you?

You would if you didn't have any ability to observe those sizes and orientations.

> thinking that there’s anything that exists

Not from “that half of something had a value”, but from “that half of any thing has a value”.

If you accept that every natural number has a successor which is a natural number, and no two natural numbers have the same successor, and that there’s no loops (e.g. by saying that there’s a total order on natural numbers and that any natural number is less than its successor), then there can’t be a finite collection which is all the natural numbers.

You could say “there’s no collection which has all the natural numbers”, which, ok, how do you want to talk about things true of all natural numbers then?

Formulating descriptions of physics without the axiom of infinity (or, without something to play the role of the real numbers) is super icky. You, in practice, can’t do any significant mathematical physics in an ultrafinitistic approach.

> how do you want to talk about things true of all natural numbers then

There's an entire branch of math for that: https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_...

I’m aware of constructive math. You still have the type of natural numbers in that?

Huh? I thought color confinement prevented this?

I think the issue might be that some people don’t actually mean “every” when they say “every”, and don’t recognize when they are speaking hyperbolically?

Or, something like that?

Yes, That plus a tendency toward binary thinking, which is something many people on hn seem to heavily suffer from.

Meteorologist are fairly accurate. People have a bias to remember more the times they were wrong.

Which logic are you saying “can’t encode the speculative moment”?

I think the two logics can emulate one another? Or, at the very least, can describe what the other concludes. I know intuitionistic logic can have classical logic embedded in it through some sort of “put double negation on everything”. I think if you add some sort of modal operator to classical logic you could probably emulate intuitionistic logic in a similar way?

You don't even need to add a modal operator since modal logic itself can be embedded in classical logic via possible-world semantics. Of course the whole thing becomes a bit clunky - but that's the argument for starting with intuitionistic logic, where you wouldn't need to do that.

Any logic with LEM

This isn’t quite right. Classical logic doesn’t permit going from “it is impossible to disprove” to “true”. For example, the continuum hypothesis cannot be disproven in ZFC (which is formulated in classical logic (the axiom of choice implies the law of the excluded middle)), but that doesn’t let us conclude that the continuum hypothesis is true.

Rather, in classical logic, if you can show that a statement being false would imply a contradiction, you can conclude that the statement is true.

In intuitionistic logic, you would only conclude that the statement is not false.

And, I’m not sure identifying “true” with “provable” in intuitionistic logic is entirely right either?

In intuitionistic logic, you only have a proof if you have a constructive proof.

But, like, that doesn’t mean that if you don’t have a constructive proof, that the statement is therefore not true?

If a statement is independent of your axioms when using classical logic, it is also independent of your axioms when using intuitionistic logic, as intuitionistic logic has a subset of the allowed inference rules.

If a statement is independent, then there is no proof of it, and there is no proof of its negation. If a proposition being true was the same thing as there being a proof of it, then a proposition that is independent would be not true, and its negation would also be not true. So, it would be both not true and not false, and these together yield a contradiction.

Intuitionistic logic only lets you conclude that a proposition is true if you have a constructive/intuitionistic proof of it. It doesn’t say that a proposition for which there is no proof, is therefore not true.

As a core example of this, in intuitionistic logic, one doesn’t have the LEM, but, one certainly doesn’t have that the LEM is false. In fact, one has that the LEM isn’t false.

People keep saying this, but the only ways I know of for formalizing this statement, appear to be probably false?

I don’t know what this claim is supposed to mean.

If it isn’t supposed to have a precise technical meaning, why is it using the word “interpolate”?

simplifier has a 4x4 version : https://simplifier.neocities.org/4x4