Philosophy Now Forum

Skepdick wrote: ↑Mon Nov 18, 2024 12:10 pm

godelian wrote: ↑Mon Nov 18, 2024 12:09 pm "Problematic" is not a standard predicate in mathematics. Not sure how it is supposed to be implemented.

It's a judgment. No different to "true".

Seeming as you've been using it throughout this exchange why can't you encode a predicate for it?

The truth predicate has a precise definition. Any predicate φ is the truth predicate if the following is true: ∀ s ( s ⟺ φ(⌜s⌝)) with s any sentence. So, it is not just a judgment.

godelian wrote: ↑Mon Nov 18, 2024 1:05 pm The truth predicate has a precise definition. Any predicate φ is the truth predicate if the following is true: ∀ s ( s ⟺ φ(⌜s⌝)) with s any sentence. So, it is not just a judgment.

Thesymbol "⟺" expresses a judgmental equality. You are trapped in equational reasoning.

https://ncatlab.org/nlab/show/equality# ... l_equality

You are judging things as "equal"; or "problematic".

Reply to immanuel Can from Belinda:

Sorry not time right now to reply to the whole of your interesting post but here is my reply regarding the nature of superstition.

Superstition is attempting to influence fate, God, or gods by means of repeating a religious or a secular ritual.

Belinda wrote: ↑Mon Nov 18, 2024 7:24 pm Reply to immanuel Can from Belinda:

Sorry not time right now to reply to the whole of your interesting post but here is my reply regarding the nature of superstition.

Superstition is attempting to influence fate, God, or gods by means of repeating a religious or a secular ritual.

Well, that's certainly not the common definition, I think we'd have to say. It would mean that astrology, belief in luck, and belief in ghosts weren't "superstitions," because they don't involve any "repeated ritual" or "attempt to influence the gods or fate".

Skepdick wrote: ↑Mon Nov 18, 2024 2:52 pm The symbol "⟺" expresses a judgmental equality. You are trapped in equational reasoning.
https://ncatlab.org/nlab/show/equality# ... l_equality
You are judging things as "equal"; or "problematic".

Well, I guess I mostly use judgmental/propositional equality, but somewhere in my hind brain I am certainly aware of the fact that this is potentially problematic. Even when dealing with Javascript, you eventually end up running into issues related to "typal equality":

( a === b ) ⟺ ( a == b && typeof(a) == typeof(b) )

But then again, the nLab page loses me when it goes on elaborating on profound results in Martin-Löf dependent type theory. I have never really dug deep enough into advanced type theory to understand the argument because I never really needed to. How related is mathematical type theory to the type systems, static and dynamic, that we ordinarily use in software? Are these things related at all? Any idea?

The biggest issue I have run into concerning the notion of mathematical equality, is what Victoria Gitman writes in her lecture on models of arithmetic:

https://victoriagitman.github.io/talks/ ... metic.html

In particular, a nonstandard model of arithmetic can have indiscernible numbers that share all the same properties.

Are these indiscernible numbers "equal" or not?

godelian wrote: ↑Tue Nov 19, 2024 4:18 am How related is mathematical type theory to the type systems, static and dynamic, that we ordinarily use in software? Are these things related at all? Any idea?

It's precisely because I have intuitive understanding of software engineering/distributed systems is why I don't care about 1st order logic; and why type theoretic foundations make more sense to me than "classical mathematics".

I wasn't indoctrinated. My understanding emerged organically.

godelian wrote: ↑Tue Nov 19, 2024 4:18 am In particular, a nonstandard model of arithmetic can have indiscernible numbers that share all the same properties.

That's liebnitz's law: https://en.wikipedia.org/wiki/Identity_ ... scernibles

godelian wrote: ↑Tue Nov 19, 2024 4:18 am Are these indiscernible numbers "equal" or not?

Yes. No. Neither. It's a definitional choice.

Consider the axiom: for all x,y in set S: x = y

From the axiom you can prove that this satisfies the definition of sub-singleton: a set that's either empty or a singleton.

And just like that I have a binary number system; and averything I know about computation applies without any formal training in Mathematics.
This is only true in Classical Mathematics with excluded middle. Without excluded middle you can't definitely say whether the set is empty or not.

And just like that I have a qubit - a set in superposition. It's both empty and non-empty. And from here-on you can can troll Classical logicians all you want.

Because that's a contradiction.

Skepdick wrote: ↑Tue Nov 19, 2024 6:59 am

godelian wrote: ↑Tue Nov 19, 2024 4:18 am How related is mathematical type theory to the type systems, static and dynamic, that we ordinarily use in software? Are these things related at all? Any idea?

It's precisely because I have intuitive understanding of software engineering/distributed systems is why I don't care about 1st order logic; and why type theoretic foundations make more sense to me than "classical mathematics".

I wasn't indoctrinated. My understanding emerged organically.

I have never studied model theory at university either.

I just came across the field by doing some disaster tourism in the foundational crisis of mathematics. It is just a hobby of mine. I'd love to book a tour of Chernobyl reactor number four, and to admire how it all went wrong. I find it incredibly fascinating. The foundational crisis of mathematics profoundly informs my worldview in terms of metaphysics.

If I tossed first-order logic out of the window, I would have to throw away a big chunk of the literature on model theory. I don't want to do that. It would become an impediment to continuing my hobby of disaster tourism.

godelian wrote: ↑Tue Nov 19, 2024 7:10 am If I tossed first-order logic out of the window, I would have to throw away a big chunk of the literature on model theory. I don't want to do that. It would become an impediment to continuing my hobby of disaster tourism.

Or you could just treat order as parametric.

order(n). If you want n=1 so be it. But it could be anything.

Classical Mathematics is worthless to software engineers. It's the very essence of our paradigm to construct the deciders.
Mathematicians just take it as a given (oh! it exists). Yeah? Write the code then!
Classical Mathemaics is "all existence, no construction"

Is the sub-singleton (as defined in first-order logic: for all x,y in set S: x = y) empty or not?
The standard (worthless) response is: f(x) = {1 if S is non-empty, 0 if S is empty}.
Great, why not g(x) = {1 if S is empty, and 0 if S is non-empty} ?
That's just re-stating the question in fuctional notation without actually answering it.

Abstract definitions don't translate into working code.

But if you really really want a theory... Closed Monoidal Categories give you a formal treatment of the Either() monad which is precisely the computational structure needed to decide the sub-singleton case by giving you a computational structire that proves LEM.

Insofar as Mathematics is just the manipulation of encodings/notation - it has uncountably many foundations.

If you want expressive power (and you do!) this is the foundational system that won't fuck up your hard-earned intuitions: https://homotopytypetheory.org/book/

It's a way of thinking which empowers you to think in terms of structure and relationships. Free from the silly notion of "identity". Free from denotation/representation.

Suppose I tweak the axiom to: for all x, in multiset S: x = y.
And just like that you get multiplicity without identity/ordering.

Does S contain numbers? Real numbers? Other kinds of objects? Don't care!
Is it ordered? No.
What's its cardinality? Don't know!

Skepdick,

I am educated in philosophy and religion, I have the credit hours and degree. I rarely say this to anyone but, “You do not have the education or intellect to understand what I am talking about.” All you do is copy and paste, then pawn it off as your own. You want to act like a skeptic? Your references are a joke, and make me snicker at your educational level. You are a little brat, and nothing more, you do not even understand what a skeptic is and how it works.

puto wrote: ↑Tue Nov 19, 2024 9:39 am

Skepdick,

I am educated in philosophy and religion, I have the credit hours and degree. I rarely say this to anyone but, “You do not have the education or intellect to understand what I am talking about.” All you do is copy and paste, then pawn it off as your own. You want to act like a skeptic? Your references are a joke, and make me snicker at your educational level. You are a little brat, and nothing more, you do not even understand what a skeptic is and how it works.

Educated? You mean indoctrinated.

Is there even an original thought in your brain; or did you have to pay in order to fill your mind with the thoughts of others?

It doesn't take much to see you are the sort of clown who hides their inability to think for themselves behind the eppaulettes of education.

Military folk have an apt term for people like you: 303. Three stars on one shoulder. Nothing. Three stars on the other shoulder.

Skepdick wrote: ↑Tue Nov 19, 2024 7:17 am Classical Mathematics is worthless to software engineers.
Classical Mathemaics is "all existence, no construction"

In my own opinion, to an important extent, this is true.
However, it sometimes still manages to reach fascinating conclusions.

Skepdick wrote: ↑Tue Nov 19, 2024 7:17 am Insofar as Mathematics is just the manipulation of encodings/notation - it has uncountably many foundations.

Well, large fragments of every foundational theory seems to be bi-interpretable with every other foundational theory. Yet another foundation will have to bring something new to the table. Otherwise, it will be just a redundant addition for something we have already.

Immanuel Can wrote: ↑Mon Nov 18, 2024 7:45 pm

Belinda wrote: ↑Mon Nov 18, 2024 7:24 pm Reply to immanuel Can from Belinda:

Sorry not time right now to reply to the whole of your interesting post but here is my reply regarding the nature of superstition.

Superstition is attempting to influence fate, God, or gods by means of repeating a religious or a secular ritual.

Well, that's certainly not the common definition, I think we'd have to say. It would mean that astrology, belief in luck, and belief in ghosts weren't "superstitions," because they don't involve any "repeated ritual" or "attempt to influence the gods or fate".

Yes, most people would say those are superstitions. However belief in luck is bad logic based on the Monte Carlo Fallacy. Belief in astrology is antiquated science.Belief in ghosts is due to hallucinations and bad science.
Belief that one can magic the gods or God to do what one wants them to do or to propitiate them interferes with proper way to relate to God.

Belinda wrote: ↑Tue Nov 19, 2024 5:20 pm

Immanuel Can wrote: ↑Mon Nov 18, 2024 7:45 pm

Belinda wrote: ↑Mon Nov 18, 2024 7:24 pm Reply to immanuel Can from Belinda:

Sorry not time right now to reply to the whole of your interesting post but here is my reply regarding the nature of superstition.

Superstition is attempting to influence fate, God, or gods by means of repeating a religious or a secular ritual.

Well, that's certainly not the common definition, I think we'd have to say. It would mean that astrology, belief in luck, and belief in ghosts weren't "superstitions," because they don't involve any "repeated ritual" or "attempt to influence the gods or fate".

Yes, most people would say those are superstitions. However belief in luck is bad logic based on the Monte Carlo Fallacy. Belief in astrology is antiquated science.Belief in ghosts is due to hallucinations and bad science.

So...what you are calling "superstition" is not as large a definition as most people would accept. Most would say that, say, somebody who refuses to walk under ladders is "superstitious," not "a victim of bad science."

Belief that one can magic the gods or God to do what one wants them to do or to propitiate them interferes with proper way to relate to God.

Well, please tell me what you know about what you are calling a "proper" way to relate to God. What makes it "proper," and how do you know?

Immanuel Can wrote:
Well, please tell me what you know about what you are calling a "proper" way to relate to God. What makes it "proper," and how do you know?

Belinda replies:
It's common knowledge that most people in the UK and Europe generally don't have faith in a supernatural way of being, nor miracles.
Christianity is a good and great religion because a man who is a also a movable icon is its paradigm of goodness.
In order for Xianity to retain any power for good in today's world we need to re-interpret it. "A proper way to relate to God" is the existentialist way. There is no essence that is God but we may create the spirit of good, truth, and beauty by our words and acts.Indeed, without man's creating spirit there could be no Christianity.

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