All expressions are composed of grades of truth considering all assertions, baring those which contradict (yet the contradiction can be composed of true assertions thus even a contradiction has a truth value), are true through a recursion where the same thing is expressed in multiple ways.
For example 2+2=4 is true as 2+2 observes 4 as a variation of 2+2. However 2+2=5 is a contradiction for the very same reason, yet the assertions which 2+2=5 is composed of still express a recursive self referentiality under 1+1=2 and 1+1+1+1+1=5.
2+2=5
Re: 2+2=5
This comment wins the irony high score.
It is infact Turing-computable (formally provable) that there are very many Mathematical theories in which 2+2=5
https://twitter.com/andrejbauer/status/ ... BXVmhskaQA
Re: 2+2=5
Unless you have a plan to rewrite the sciences of arithmetic and set theory, I can only assume that you plan to redefine the natural number line as 1,2,3,5,4,6...
This of course is theoretically allowable, provided you redefine the number "5" to mean what has been traditionally regarded as the meaning of the number "4". Or do you have another plan? Please share.
This of course is theoretically allowable, provided you redefine the number "5" to mean what has been traditionally regarded as the meaning of the number "4". Or do you have another plan? Please share.
Re: 2+2=5
They aren't sciences. They are conventions.
Whether the natural numbers start at 0, 1 or -56351231 is arbitrary choice.
Numbers (on their own) have no meaning. Except in relation to all other numbers.
The most trivial way to convince yourself of this fact is to attempt to tell me something objective about the number 5. A truth that is not a relation to other numbers/mathematical objects.
Or if you want to go full tilt - numbers don't exist.
Last edited by Skepdick on Sun Feb 06, 2022 10:29 am, edited 1 time in total.
Re: 2+2=5
I am not appealing to Twitter?
I am citing a formal proof by a professor of computational mathematics. Twitter just happens to be where he published his work.
You can reproduce the result for yourself using the source code: https://github.com/andrejbauer/alg
Science!
Re: 2+2=5
Andrej Bauer is a great mathematician and terrific expositor. I haven't read through this thread, but I sincerely doubt he holds that 2 + 2 = 5.
ps -- He's talking about commutative unital semirings of size at most 6. If you can define that without looking it up I'll take your point, else you have posted Bauer's comment wildly out of context.
Re: 2+2=5
On what grounds/evidence do you doubt this?
http://math.andrej.com/2012/10/03/am-i- ... ematician/
But seriously, if anything, you may call me a mathematical relativist: there are many worlds of mathematics, and the view of the worlds is relative to which one I am in. Any attempt to bring mathematics within the scope of a single foundation necessarily limits mathematics in unacceptable ways. A mathematician who sticks to just one mathematical world (probably because of his education) is a bit like a geometer who only knows Euclidean geometry. This holds equally well for classical mathematicians, who are not willing to give up their precious law of excluded middle, and for Bishop-style mathematicians, who pursue the noble cause of not opposing anyone.
What could be more appealing to a mathematician than the idea that there is not one, but many, infinitely many worlds of mathematics? Would he not want to visit them all, understand how they are related, and see what happens to his favorite subject as he moves between them?
It's crazy ironic and wildly hypocrytical that you are insisting "2+2=5" is "out of context" while forgetting to make the context you have in mind explicit.
Mathematical Universes. Contexts. What's the difference?
It's also weird that you don't recognise that just about any formal expression can be promoted to a theorem. And then we can proceed the conversation in the knowledge that we are talking about ALL the Universes in which the theorem holds.
If I choose to evaluate the expression "2+2" in the Universe of unital semirings of size at most 6, then I will damn well evaluate it that Universe.
If I choose to evaluate the expression "2+2" in the Universe of some higher-order structures of my own invention that have no pre-existing Mathematical denotation (or a denotation that I care not to learn; or remember) then I will damn well evaluate it in that Universe.
It's so peculiar that Mathematicians don't understand the implications of the is-ought gap. By telling me that I am taking something "out of context" you implicitly hold some range of Universes (or perhaps even some particular universe) as being THE "correct" Universe for evaluating "2+2".
And since you have failed to make your Universe explicit, and you have failed to justify why your Universe is the one in which we ought to evaluate "2+2" then I should be able to accuse you; and successfully convict you of intellectual dishonesty.
Frankly, I think you just want to have your cake and eat it too. You want the Mathematical Multiverse to be purely abstract and infinitely unconstrained - you want to live in Cantor's paradise, while you also want to reject 2+2=5 by rejecting all the Universes in which it's true.
Re: 2+2=5
Chaps, I think we are getting a little overheated! The various branches of mathematics - arithmetic, set theory, geometry etc - are well-established sciences based firmly upon agreed axiom sets. It's perfectly acceptable, indeed desirable, to experiment by arbitrarily varying the axioms; this is, after all, how non-Euclidean geometry got started. But this should only be done for sound reasons. So long as we are debating a question in conventional arithmetic, for example, we do well to stick to the received axiom set. One who asserts that "I believe that 2+2 might equal 5, because I choose to disregard the Principle of Induction", really leaves no common ground for the rest of us to engage discussion.
Re: 2+2=5
That's called a bandwagon fallacy.
Precisely. What is the objective decision procedure, the arbiter of "sound reasons"?
Or do you have some pre-agreed axiom set of sound reasons?
Appeal to authority.
Strawman argument. I don't disregard the principle of induction.
ALL Mathematical proofs are inductive types!
Which is precisely in accordance with my axiom.
Views of the Mathematical multiverse are relative to where each Mathematician stands, and they are irreconcilable.
Of course this is also a direct implication of the leader election problem in Computer Science.
Which axiom sets should we agree to? Every Mathematician believes that we should agree to their axiom-set!
The problem is undecidable.
Last edited by Skepdick on Tue Feb 08, 2022 12:05 pm, edited 2 times in total.
Re: 2+2=5
Success!!!
Now, it's up to you to determine who should terrify you more.
Somebody that passes your Turing test; or somebody that intentionally fails it.
P != NP
Did I also mention... fuck axiomatic reasoning? And in general - fuck all forms of fundamentalism.
Re: 2+2=5
Like I say, if you can't define it without looking it up, then you're full of beans.
ps -- Based on our previous conversations, I suspect you couldn't define it even if you did look it up.
pps -- Do you know what a ring is? A semiring? A unital semiring? Can you give some examples of each? If not, then who exactly are you trying to impress by name checking things you know nothing about?