The Countable (Dedekind) Reals
Posted: Thu May 12, 2022 6:07 pm
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The Countable (Dedekind) Reals
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Re: The Countable (Dedekind) Reals
Posted: Sat May 14, 2022 2:03 pm
Not interested in what someone on YouTube has to say. This is supposed to be a philosophy forum; summarise the argument in your own words.
Re: The Countable (Dedekind) Reals
Posted: Sun May 15, 2022 1:00 am
To be fair, Andrej Bauer is a famous constructive mathematician and a highly lucid and interesting expositor. The problem here is that the OP knows how to post a link, but has no actual understanding of the video himself.
Re: The Countable (Dedekind) Reals
Posted: Mon May 16, 2022 9:16 am
The summary of the argument is in the subject line. The Real numbers are countable (with the fine print being if you define everything just the right way).
The moral of the story is that Mathematics is invented, not discovered.
Because it can't be true that both the Reals have the universal property of countability, and the reals lack the universal property of countability.
The "universal property" (what a bullshit phrase) of countability in the Reals is subject to your choice of topos.
Re: The Countable (Dedekind) Reals
Posted: Mon May 16, 2022 9:21 am
The problem with commentary like this is that the commenter doesn't even understand what it means to understand. The notion of understanding is not well-defined, yet he asserts that he posesses it while others don't. It's a bit like asserting you have consciousness - it means whatever you want it to mean.
Part and parcel of your problem is that the whole point of intuitionism is that there is nothing to understand. Every concept in Mathematics is invented by virtue of toposes being simply contexts in the internal logic of the inventor's head. By virtue of invention every concept is at best a tautology of the inventor's desires.
And Andrej Bauer himself demonstrates that if he wants the Reals to posess some universal property such as "countability" then he shall invent precisely the topos (CONTEXT!) he needs; he accepts precisely the axioms he needs to accept; and rejects precisely the axioms he needs to reject to satisfy the universal property he desires.
You have your head burried so deep up your abstract Mathematical ass that you still don't understand that there's no difference between object construction and programming. Peter Naur pointed this out in 1985.
What ivory tower Category Theorists call "universal properties" are exactly the same thing as what run-of-the-mill software engineers call interfaces.
Definition is reification.
Re: The Countable (Dedekind) Reals
Posted: Mon May 16, 2022 3:10 pm
Mathematicians devised methods for solving problems. Those methods were always available waiting to be discovered.Skepdick wrote: ↑Mon May 16, 2022 9:21 amThe problem with commentary like this is that the commenter doesn't even understand what it means to understand. The notion of understanding is not well-defined, yet he asserts that he posesses it while others don't. It's a bit like asserting you have consciousness - it means whatever you want it to mean.
Part and parcel of your problem is that the whole point of intuitionism is that there is nothing to understand. Every concept in Mathematics is invented by virtue of toposes being simply contexts in the internal logic of the inventor's head. By virtue of invention every concept is at best a tautology of the inventor's desires.
And Andrej Bauer himself demonstrates that if he wants the Reals to posess some universal property such as "countability" then he shall invent precisely the topos (CONTEXT!) he needs; he accepts precisely the axioms he needs to accept; and rejects precisely the axioms he needs to reject to satisfy the universal property he desires.
You have your head burried so deep up your abstract Mathematical ass that you still don't understand that there's no difference between object construction and programming. Peter Naur pointed this out in 1985.
What ivory tower Category Theorists call "universal properties" are exactly the same thing as what run-of-the-mill software engineers call interfaces.
Definition is reification.
Discovered doesn't mean invented.
Re: The Countable (Dedekind) Reals
Posted: Mon May 16, 2022 3:16 pm
Words, words, words. Meaningless words.jayjacobus wrote: ↑Mon May 16, 2022 3:10 pm Mathematicians devised methods for solving problems. Those methods were always available waiting to be discovered.
Discovered doesn't mean invented.
If one claims to subscribe to the axiom of non-contradiction (as Mathematicians claim that they do!) it is not possible that:
(ℝ is countable) ∧ (ℝ is not countable) ⇔ True
Re: The Countable (Dedekind) Reals
Posted: Mon May 16, 2022 3:25 pm
My words have meaning which you can look up in a dictionary.Skepdick wrote: ↑Mon May 16, 2022 3:16 pmWords, words, words. Meaningless words.jayjacobus wrote: ↑Mon May 16, 2022 3:10 pm Mathematicians devised methods for solving problems. Those methods were always available waiting to be discovered.
Discovered doesn't mean invented.
If one claims to subscribe to the axiom of non-contradiction (as Mathematicians claim that they do!) it is not possible that:
(ℝ is countable) ∧ ¬(ℝ is countable) ⇔ True
Your words have meaning but I don't know what you are trying to say.
Re: The Countable (Dedekind) Reals
Posted: Mon May 16, 2022 3:27 pm
The meaning of your words comes from a dictionary? Shame.jayjacobus wrote: ↑Mon May 16, 2022 3:25 pm My words have meaning which you can look up in a dictionary.
I use my words as necessary.
Which part of this is confusing you?jayjacobus wrote: ↑Mon May 16, 2022 3:25 pm Your words have meaning but I don't know what you are trying to say.
IF one takes the axiom of non-contradiction seriously it is cannot be true that...
ℝ is countable.
AND
ℝ is not countable.
That's a contradiction.Re: The Countable (Dedekind) Reals
Posted: Mon May 16, 2022 3:37 pm
So don't say that.Skepdick wrote: ↑Mon May 16, 2022 3:27 pmThe meaning of your words comes from a dictionary? Shame.jayjacobus wrote: ↑Mon May 16, 2022 3:25 pm My words have meaning which you can look up in a dictionary.
I use my words as necessary.
Which part of this is confusing you?jayjacobus wrote: ↑Mon May 16, 2022 3:25 pm Your words have meaning but I don't know what you are trying to say.
IF one takes the axiom of non-contradiction seriously it is cannot be true that...
ℝ is countable.
AND
ℝ is not countable.
That's a contradiction.
Re: The Countable (Dedekind) Reals
Posted: Mon May 16, 2022 3:38 pm
Why not? I want to know!
Is ℝ countable; or is ℝ not countable?
Google and the status quo of Mathematics are telling me ℝ is not countable.
I have published and peer-reviewed Computer Scientist telling me ℝ is countable.
Who is lying to me? Who do I believe?!?
Re: The Countable (Dedekind) Reals
Posted: Mon May 16, 2022 3:45 pm
Believe who you want. I don't believe someone who contradicts himself.Skepdick wrote: ↑Mon May 16, 2022 3:38 pmWhy not? I want to know!
Is ℝ countable; or is ℝ not countable?
Google and the status quo of Mathematics are telling me ℝ is not countable.
I have published and peer-reviewed Computer Scientist telling me ℝ is countable.
Who is lying to me? Who do I believe?!?
Re: The Countable (Dedekind) Reals
Posted: Mon May 16, 2022 3:47 pm
That's fucking useless advice.jayjacobus wrote: ↑Mon May 16, 2022 3:45 pm Believe who you want. I don't believe someone who contradicts himself.
I want to believe true things. I don't want to believe false things. So who is telling the truth; and who is lying?
Some experts are saying ℝ is countable.
Other experts are saying ℝ is not countable.
Who should we believe when the "experts" (who keep reminding us we "have no actual understanding") are contradicting each other?!?
Re: The Countable (Dedekind) Reals
Posted: Mon May 16, 2022 5:13 pm
Catch a tiger by the toe .........Skepdick wrote: ↑Mon May 16, 2022 3:47 pmThat's fucking useless advice.jayjacobus wrote: ↑Mon May 16, 2022 3:45 pm Believe who you want. I don't believe someone who contradicts himself.
I want to believe true things. I don't want to believe false things. So who is telling the truth; and who is lying?
Some experts are saying ℝ is countable.
Other experts are saying ℝ is not countable.
Who should we believe when the "experts" (who keep reminding us we "have no actual understanding") are contradicting each other?!?
Re: The Countable (Dedekind) Reals
Posted: Mon May 16, 2022 5:18 pm
Uhuh... so Mathematical truths are subject to arbitrary choices?!?!