The CPU would simple directly input integer values into the OS by having mounted on the RAM and bypassing slow hard drives through an application like TimeShift. — Shawn
fishfry, may I ask for your opinion? — Shawn
utilizing the ECC-RAM utilized nowadays in servers to be able to predetermine the state of a computer through directly interacting with the system OS itself. — Shawn
I get a lot if insights from your posts on maths. — Wayfarer
But then, the person that might end up paying the price for that might be someone they infect. So perhaps the compromise is, anyone has a right to refuse to be vaccinated, but by so doing they forfeit the right to move freely in society. — Wayfarer
Smart men never get married lol. It's a bad deal. — hope
not von Neumann but of neural networks — D2OTSSUMMERBUG
Where does the creativity of AI come from? — D2OTSSUMMERBUG
So … can we say that this rule is where all morality should stem from? — Deus
I begin this thread in response to the backlash I received — Leghorn
In fine, courage used to be overcoming fear. Now it is succumbing to it. — Leghorn
Is there anything wrong in stating that neurotransmitters are scientifically assumed to play a role in the regulation and experience of affective behavior? — Shawn
Antidepressants are supposed to make people feel happier and more at ease, but a study has linked several prescription antidepressants to an increased risk of violent behavior, including physical assault and homicide.
The first video (I didn't watch the second video) is stupid nonsense and disinformation.
In this context, infinite summation is defined only for converging sequences. If the rules of definition are violated, then, of course, contradictions may be derived. There is no mystery or even problem about that.
The person at the blackboard says, "The problem with infinity is all sorts of weird things happen when you're dealing with infinity". First, that doesn't even mean anything. Second, instead of explaining that the fallacy is in using an undefined notion (infinite summation on a sequence that does not converge), the person at the blackboard doesn't even suggest how we may investigate further to see that there is not an actual conundrum.
The video is yet another example of Internet ignorance and disinformation. That person seems to be teaching a classroom. He should be told by the school administrators to clean up his act: If he wants to present mathematical challenges, then he should provide his students with the benefit of techniques and information for solving the challenges rather than obfuscate with "weird things happen". — TonesInDeepFreeze
Contra that, in the item I cited, there are mentions of empiricism and the example of the applicability of the mathematics of elasticity to engineering a bridge; see (3) here. — Banno
so again:
It seems that for — Banno
, being consistent and being true are inseparable; that mathematical truth is displayed in mathematic's empirical applicability; and that hence the consistency of mathematics shows that it is empirically applicable and hence incompatible with the syntactic view.
— Banno
So if we trust this secondary source - I have no reason not to - Gödel held something like that we can speak of mathematical propositions being true because they are empirically applicable; — Banno
that truth is at a meta-level to the mathematics itself; and that together these show that maths is not just stuff we make up. — Banno
The exact argument remains opaque, but that is the implication. — Banno
From what I could glean from Wikipedia, a constructible set is one which can be, well, constructed via set theoretic operations — TheMadFool
Way above my paygrade! Thanks though. I hope to advance my knowledge in math ASAP. — TheMadFool
It's trivial stuff, but may not have been around before — jgill
Probably there are different versions of him to be consider of: Platonism, realism, relativism, etc... — javi2541997
One of the reasons why some, like Gödel I suppose, believe math is discovered is how math seems to,
1. Describe nature (math models e.g. Minkowski spacetime)
2. Describe nature accurately (we can make very precise predictions to, say, the 15th decimal place) — TheMadFool
I so mean.
But we call them assumptions. — Banno
A deduction in a Hilbert-style deductive system is a list of formulas, each of which is a logical axiom, a hypothesis that has been assumed for the derivation at hand, or follows from previous formulas via a rule of inference. The logical axioms consist of several axiom schemas of logically valid formulas; these encompass a significant amount of propositional logic. The rules of inference enable the manipulation of quantifiers. Typical Hilbert-style systems have a small number of rules of inference, along with several infinite schemas of logical axioms. It is common to have only modus ponens and universal generalization as rules of inference.
Natural deduction systems resemble Hilbert-style systems in that a deduction is a finite list of formulas. However, natural deduction systems have no logical axioms; they compensate by adding additional rules of inference that can be used to manipulate the logical connectives in formulas in the proof.
Quick question: why can't we throw out self reference with regard to Gödel like they did with Russell's paradox? — Gregory
Self-reference has a problem, if you want to think about it in terms of "I am not provable" sort of approach. A well-formed formula cannot refer to itself. Moreover, a formula cannot refer to the meta-theory (which is where proofs exist).
What Gödel did was two things:
1. Internalize the meta-theory into the natural numbers via coding, and show that this internalization is very robust.
2. Showed that there is a sentence with Gödel number n, whose content is exactly "the sentence coded by n is not provable".
The importance is in both points. They allow us both (limited) access to the meta-theory and the proofs; as well circumvent the problem of being a well-formed formula while still referring to itself. And while the importance of the incompleteness theorem is mainly in the fact that it shows there is no reasonable way to have a finitary proof-verification process to mathematics, and also prove or disprove every sentence; the proof itself is also important because it gives us the internalization of the meta-theory into the natural numbers.
No axioms. That's pivotal. Instead there is a rule of assumption: one can introduce any proposition on chooses at any time, and rules for deriving more theorems from those assumptions — Banno
SO the lesson might be that when the love of axioms tried to tighten up mathematics, it ended up toppling the axioms. — Banno
I suspect that in mathematics any true formula can serve as an axiom form which to develop more cool mathematics. That's quite a different thing to an axiom in logic. — Banno
Need to take a break so we don't post over each other. Yes, ND is just deduction. But in ND, any theorem can be taken as axiomatic, to be discharged as the deduction proceeds. As I say, they are functionally equivalent. — Banno
Too fast. Why? — Banno
What can you assume in natural deduction?
In natural deduction, to prove an implication of the form P ⇒ Q, we assume P, then reason under that assumption to try to derive Q. If we are successful, then we can conclude that P ⇒ Q. In a proof, we are always allowed to introduce a new assumption P, then reason under that assumption.
But I can see how we might have different pictures of mathematics, which are despite that functionally equivalent: one of maths as a series of axiomatic systems, the other as a language that becomes increasingly complex as new formation rules are added. — Banno
Is there a reason to think that natural deduction is not as powerful as axiomatisation? — Banno
Hmm. Your puzzlement has me puzzled, — Banno
Or we might treat it as a series of rules for derivation, never complete but always consistent, to which we add new rules of derivation in a similarly asymptote fashion. — Banno
Did you use an axiomatic system? — Banno
I studied logic forty years ago. — Banno
Of course. I'm saying don't bother with axioms. — Banno
I intentionally left axioms out of my post. I've always been struck by the fact that what we select as our axioms is more or less conventional; we might have chosen different axioms. So in that way if one supposes that "all mathematical truth is derived from axioms" all one is doing is insisting on coherence and completeness. — Banno
SO Gödel is clarifying the syntactic view while rejecting it. — Banno
This is a vast topic. The more I look in to it the more it grows. And it is difficult. — Banno
I'll come out straight up and say that my own prejudice is that maths is made up as we go along, — Banno
In the SEP article I cited above, an account is given of an argument from Gödel against intuitionist and constructivist views, which Gödel grouped as the "syntactic view", such that the truth of mathematical propositions is determined by their relation to each other, by their syntax, and not by their dependence on facts. — Banno
It seems that for Gödel, being consistent and being true are inseparable; that mathematical truth is displayed in mathematic's empirical applicability; and that hence the consistency of mathematics shows that it is empirically applicable and hence incompatible with the syntactic view. — Banno
The argument is unclear — Banno
places Gödel in the position of having to choose between vicious circles and "classical" mathematics. Gödel rejects vicious circles. — Banno
So back to my own musings. I'd be interested to see if paraconsistent mathematics might present a distinction between consistency and truth that would mitigate against Gödel's almost equating the two. This would count against his argument in the SEP article. — Banno
I'd also be interested in work that sets out how mathematical rules might be shown (a philosophically loaded term after Wittgenstein) in empirical situations, rather than expressed in mathematical terms. See The Epistemology of Visual Thinking in Mathematics — Banno
It needs to be borne in mind that mathematics is ultimately a system, game-like in nature, where we have complete freedom to choose our starting premises aka axioms. — TheMadFool
Exactly in this part, I guess Gödel and Ravitch concided. Those axioms or propositions are so related with the "realism" itself. — javi2541997
Still, climate change is more of a "when", as opposed to nuclear armageddon's "if". — hypericin
Reading deeply the dissertation ..., — javi2541997
Gödel has defended classical mathematics against each of the major programs of restricted methods. He has rejected intuitionism, semi-intuitionism, the vicious circle principle, and constructive or finitist programs in general. has defended classical mathematics against each of the major programs of restricted methods. He has rejected intuitionism, semi-intuitionism, the vicious circle principle, and constructive or finitist programs in general.
That sounds like a good topic for a new thread. — Metaphysician Undercover
Yes; that's what I said. — Banno
The argument seems to be that a system's being true (about bridges) is on a par with its being consistent, in that we can only show either by bringing in something form outside - this being a consequence of the second incompleteness theorem.
I'd be interested to hear if others think this an accurate account of Gödel's thinking. — Banno
The argument seems to be that a system's being true (about bridges) is on a par with its being consistent, in that we can only show either by bringing in something form outside - this being a consequence of the second incompleteness theorem. — Banno
I'd be interested to hear if others think this an accurate account of Gödel's thinking. — Banno
Then how is it ok to impose this situation on any child, let alone your own? — hypericin
Seconded. I have a distinct recollection of having written something intelligent. Hate to lose 'em - doesn't happen all that often. — tim wood
The thing with this type of deception — Metaphysician Undercover
In other words "If 2 + 2 = 5 then I am the Pope" is a true material implication. Do you understand that? Do you agree? Do you have a disagreement perhaps?
— fishfry
Sorry fishfry, but you'll need to do a better job explaining than this. Your truth table does not show me how you draw this conclusion. — Metaphysician Undercover