that fact will hasten the end of the fossil fuel era, due to the fact that fossils are inherently inferior to electric. [...] It's a matter of simple economics with climate change. A cost-benefit analysis is all that is needed to persuade a politician, and solar and the wind is becoming so cheap in many regions, that people are seriously considering the switch. — Posty McPostface

That is precisely what Peirce and I mean by "hypothetical states of affairs" as the subject matter of pure mathematics - there is no connection (purported or otherwise) with reality as the subject matter of metaphysics. — aletheist

That depends entirely on what we mean by "doing logic." — aletheist

Charles Peirce, following his father Benjamin - one of the most accomplished American mathematicians of the 19th century - defined mathematics as the science of reasoning necessarily about hypothetical states of affairs. As such, it includes mathematical logic as distinct from and more fundamental than normative logic

I (following Peirce) hold that logic is the science of how any intelligent beings should think, if their purpose is to arrive at true beliefs by learning from experience. This is similar to the distinction that Peirce drew between logica utens, instinctive and uncritical habits of inference, and logica docens, deliberate and rigorous habits of inference. Logic has to be a normative science if there is good reasoning vs. bad reasoning, which I hope no one in this forum would deny. — aletheist

I understand that personal or corporate debt isn't the same as national debt. However, the fact is that national debt is a recurrent item of discussion among economists and politicians, both. — Bitter Crank

No, Rovelli's 'M' explicitly excludes contradiction: "Then the platonic world M is the ensemble of all theorems that follow from all (non contradictory) choices of axioms": It contains everything that is true under any choice of non-contradictory axioms (so yes, read the paper!). — StreetlightX

If you mean truths that hold in different possible worlds, then these truths constitute a more general/more abstract/higher-order possible world. — litewave

I believe that's exactly the point: M would be entirely trivial. This is the dilemma that the paper poses for Mathematical Platonism: either M is trivial and has no structure whatsoever (and thus largely says nothing at all about our world), or, if M is not trivial in this way, then it cannot be independent from our intellectual activity. In either case Platonism is undermined because if the former, then it has no explanatory power, and if the latter, then it simply isn't Platonism. — StreetlightX

part of this, in turn, has to do with the modal status of our math: contingent or necessary, and to what degree? Rovelli's answer is a kind of qualified contingency: our math is contingent ("Which tiny piece of M turns out to be interesting for us, which parts turns out to be \mathematics" is far from obvious and universal. It is largely contingent"), but this contingency in turn is premised upon the kind of beings we are, and the kind of things we encounter in the world, along with what we do with them - which lends our mathematics a kind of empirical necessity (Rovelli doesn't use that term, but I think it's appropriate in this context). — StreetlightX

National debt seems more like credit card debt to me. Some of it may be as necessary as a mortgage, but a lot of it is living beyond one's income. Now, the US Government could, if politicians were willing, increase its income through taxation, and could put a ceiling on its debt or lower its indebtedness. It would be a good thing, because the interest on the national debt is huge, and costs us the opportunity to accomplish worthwhile goals. — Bitter Crank

Private ownership, or just ownership of capital? — Marchesk

Was Marx envisioning a fully automated society? Because a post-scarcity society has never existed. — Marchesk

Now, he asks that we imagine a world M, which contains every possible mathematical object that could ever exist, even in principle. — StreetlightX

This, though, opens up a new question - what is 'interesting?' Well, interest simply is in the eye of the beholder — StreetlightX

Far from being stable and universal, our mathematics is a
fluttering buttery, which follows the fancies of inconstant creatures. Its theorems are solid, of course; but selecting what represents an interesting theorem is a highly subjective matter.... The idea that the mathematics that we find valuable forms a Platonic world fully independent from us is like the idea of an Entity that created the heavens and the earth, and happens to very much resemble my grandfather." — StreetlightX

A standard definition of Socialism is “From each according to his ability, to each according to his needs.” — tinman917

I think you have it backward. My claim is that that if there is no sufficient reason in reality, we cannot know that there is a sufficient reason. This was in response to your suggestion: — Dfpolis

Putting aside your unsupported sociological claim, yes, some people are quite irrational. How can we know there is a sufficient reason if there is no sufficient reason to know? — Dfpolis

I think that what this little exercise demonstrates, is that in some of the more politicized academic subjects anything will likely be accepted for publication in the journals, provided only that the paper's politics are perceived to be correct. Which arguably does tell us something about the academic standards of the subjects in question. — yazata

Yay you agree. I was simply pointing out that which axioms you choose to adopt cannot be determined without the use of other axioms so you ultimately end up with an arbitrary logic. — khaled

The only reason the law of identity holds as you've said is because
A) not having it would result in an incoherent and absurd system of logic and
B) a system of logic has to be coherent and consistent

My point is you cannot get A from B nor B from A and so one should just admit that they're both arbitrary because they are. — khaled

You justify A using B then claim that everyone has B. While that is true, I'm trying to find a way to get B that does not rely on consensus, pragmatism or arbitrariness (thus the title of the discussion: where does logic get its power. So far you've clearly shown that everyone has B but I'm asking WHY everyone has B and you cannot use an answer that refers to C if C is also as arbitrary as A and B) — khaled

If you allow brute facts you reject the PSR and with it the logical foundations of science. — Dfpolis

I'm not advocating triviality here. I am simply stating that you cannot explain why triviality is to be avoided without appealing to theoretical or practical uses. — khaled

Why should we have a consistent theory of mathematics? Why should we have an understanding of the natural world? Why should we seek the answers to theoretical problems? I'm not saying we shouldn't do any of these things, I'm pointing out that to have an understanding of the natural world/ to have a consistent mathematical theory, etc cannot be justified without begging the question. — khaled

You have to set these things as goals first before you discriminate against triviality/ other systems of logic. And there is nothing in classical logic that can be used to justify itself or to frown at triviality. — khaled

The statement "A=A" is not ontologically different from the statement "A!=A" and there is no proof of either statement therefore one cannot be used to justify itself or devalue the other. It's just that the people that thought A!=A died and the ones that thought A=A lived. Ultimately, logic is based on consensus between homo sapiens and there is nothing in the consensus of homo sapiens that leads one to believe a proposition is true. — khaled

That is a practical consideration. As I've said before all of your explanations as to why we should avoid triviality are practical explanations. If triviality one day proves to be a more useful form of logic we will switch to that. — khaled

All of these are practical virtues. They are virtues because they are useful. I don't mean practical as in used in physics, I mean practical as in both theoretically and physically applicable — khaled

Yes and the problem I'm having is that there is no reason for anyone to agree on assumptions that is not itself an assumption — khaled

Why not? Why should we avoid triviality? — khaled

People do use fuzzy logic in many many applications such as "facial pattern recognition, air conditioners, washing machines, vacuum cleaners, antiskid braking systems, transmission systems, control of subway systems and unmanned helicopters, knowledge-based systems for multiobjective optimization of power systems, "
Source: first site that pops up when you look up fuzzy logic uses. — khaled

Not only that, but the fact that fuzzy logic shares some axioms with classical logic does not in any way indicate that those axioms are to be shared by all systems of logic. That is a genetic fallacy. — khaled

But as for why the system SHOULDN'T blow up you've given no answer. You've simply asserted "the system should not reach the point of triviality" but you've never said why and the only reason I can think of is practical uses. — khaled

Oh really? What else plays a role? — khaled

The fact that no one disputes them is no proof of their validity. — khaled

What are the criterions of theory choice? Because as far as I know that's a matter of opinion and practical utility. One might choose to use the most elegant theory, the most accurate theory, the easiest theory to use, etc. — khaled

It IS entirely ragtag because any rule you choose to use as an axiom is by definition based on no other reasoning. — khaled

Take the Law of non Contradiction for example. In terms of practical value this law is priceless however it IS an axiom and it IS entirely arbitrary. God could've woken up one day and decided "hey you know what, let's get rid of the law of non contradiction" and created an absurd yet consistent universe. — khaled

A better example is fuzzy logic. It has no binary truth value but it is still very useful and entirely consistent. You can only say it is not ragtag to the extent that it helps us survive when applied. — khaled

Why not? This binds our logical systems to practical value, which brings it back to the definition you started your reply with. Now logic requires neither rigor not any specific axioms, it just needs to be useful when applied to the world. It just so happens that rigor is extremely useful when applied to the world so we use that in almost all logical systems — khaled

Again, you are binding logic to practical value which is exactly what the start of your comment tries to refute. — khaled

And there seems to be a pretty pragmatic explanation here. If the logic we naturally develop begins failing too often we change our logic until we find one that works. Otherwise we die so there's good incentive. — MindForged

I'm not looking for a way to justify logic in terms of practical usefulness (because you can justify almost anything that way) or in terms of consensus as a result of practical usefulness. I am looking for a way to justify it that is entirely devoid of practical uses. I think this is impossible but I wanted to see other people try. — khaled

There's two ways I can think of how to do this. You justify a deductive logic by means of abduction, a model of theory choice. Whatever logic, in some specified domain, comes out the best on the criterion of theory choice is the correct one for the domain (we can assign them scores basically). That's not question begging, it's using a different type of reasoning.

Another way would be to pick a very weak logic which contains principles no one disputes but which does not contain principles under disagreement. Whatever that logic ends up being, it would have to, for example, have a conditional which satisfies Modus Ponens. That will be a common ground across logics that are actually used. Either of these means suffice. — MindForged

If I continue the mapping a few more spots, the even numbers go to 8, 10. But these numbers will be outside of the set of natural numbers, on the left. Don't you see that with this type of mapping, the set of even numbers will always contain numbers outside the set of natural numbers which it is mapped to, so it is impossible for it to be a subset? — Metaphysician Undercover

Right, so no matter how you lay it out, if you maintain equal cardinality the set on the right side will always contain numbers which are not contained in the set on the left side. Surely you can acknowledge this. So do you agree with me that it is impossible that the set on the right side is a subset of the set on the left side? If not, why not? — Metaphysician Undercover

We both seem to have reached the conclusion that logic is: "A rule for making rules that is based off of ragtag collection of intuitions that we are born with strung together which helps us survive". You said that we are not born with logic and that we change it periodically to help us survive which I totally agree with but then that would be putting logic on the same "correctness" level as lunacy. They are both based on primordial intuitions, just that the followers of one survive and the followers of the other perish. — khaled

The problem is, there are countless potential ways to formulate such rules and there is no meta-rule about how to do this for which there are no alternatives, at least none that I can see. — khaled

The main goal for this discussion was to get people to think about such a meta-rule (A method for choosing logical axioms) for which there is no alternative. A "common ground" across all possible systems of logic if you will. Question 4 was supposed to be a trick question because what defines "work" IS the axioms of logic. You can't get an answer to "what works" without knowing something that works and you can't know what works without knowing the answer to "what works". It seems to me that the only way to develop a logical system is to pull yourself up by your bootstraps (beg the question) and I'm looking for someone to convince me otherwise

A method by which humans go from premise to premise that seems to reflect reality if the premises do. — khaled

1) What was the "origin" of logic.

2) Why is it that we are simply born with a "rule for deriving rules" and why does it work so well?

3) Why is it then that humans can get by using arbitrary axioms that they are born with whose validity they cannot prove?

4) And why is it that despite the fact that many axioms fit that description, that only very few work?

5) Again, where does logic get it's reality-reflecting power? — khaled

Is there any metaphysical basis for logic or are humans just stuck with a certain type of hardware — khaled

Clearly, your sets as written do not indicate that the right is a subset of the left. The left contains 4 and 6, which are not contained in the right. It is not a subset. — Metaphysician Undercover

Clearly, for any set of natural numbers, a proper subset is always smaller. That is always the case, and there is never an exception. — Metaphysician Undercover

the number line between 0 and 1 has length 1
- to find out how many things fit on the line
- divide line length by the thing length
- a number has length 0
- so the number of number between 0 and 1 is 1/0=UNDEFINED
- if you let number have non-zero length then there is a finate number of numbers in the interval but a potential infinity as number length tends to zero — Devans99

I can’t believe you; we’ve been talking about this for ages and you have learned nothing. You are still not even using the proper language to discuss this is (actual/potential infinity). — Devans99

You need to realise that you were told the wrong things about infinity at school and free your mind of Cantor’s muddled dogma. — Devans99

You have stated an arbitrary boundary of zero and one, but this does not bound the infinite. You could have set your boundaries as 10 and 20, or 200 and 600, or zero and the highest natural number. These boundaries do not bound the infinite itself. — Metaphysician Undercover

The boundaries are in the definitions by which they are produced, but the definitions are made such that the numbers themselves are not bounded. The two systems, the naturals and the reals, are just two distinct ways of expressing the same infinite numbers. — Metaphysician Undercover

There is a real problem with this so-called proof. It's called begging the question. By assuming that the natural numbers are a countable "set", it is implied that the naturals are not infinite. It is impossible to count that which is infinite. — Metaphysician Undercover

A computing array is obviously bounded by memory limitations as you found out when your program hung. — Devans99

The naturals {1,2,3,...} are unbounded on the right as denoted by the ...

The reals between 0 and 1 {.1, .01, .001, ... } are unbounded ‘below’.

Both are an example of potential not actual infinity in that it is an iterative process that generates an infinity of numbers.

The number of reals between 0 and 1 is undefined: a number has ‘length’ 0 and 1/0 = undefined. If you let number have length>0 you get a finite number of reals between 0 and 1. So there is no way to realise actual infinity...

I agree, but as I explained, the thing which is infinite is not the same thing as the thing which is bounded. Therefore the limits expressed are irrelevant to the infinity expressed, and the infinity is unbounded. Therefore your argument that there can be a bounded infinity is not sound. — Metaphysician Undercover

I don't believe this. Both the naturals and the reals are infinite, so I believe it is false to say that one is larger than the other. This is where I believe that set theory misleads you with a false premise. I would need some evidence, a demonstration of proof, before I would accept this, what I presently believe to be false. Show me for example, that there are more numbers between 1 and 2, and between 2 and 3, than there are natural numbers. The natural numbers are infinite. So no matter how many real numbers you claim that there are, they will always be countable by the natural numbers. — Metaphysician Undercover

This is what I've been telling you over and over again. To stipulate that the cardinality of the natural numbers is less than something else, and to also say that the natural numbers are infinite, is contradictory. — Metaphysician Undercover

"Rigorous mathematical understanding of infinity". Lol. But if your not joking, you have my sympathy. — Metaphysician Undercover

Does anyone even know what it means to be larger then zero? — Metaphysician Undercover

That there is an infinity of real numbers between any two real numbers is the assumption of infinite divisibility — Metaphysician Undercover

So the infinite thing itself, divisibility, is not bounded. Likewise, in the case of the natural numbers, that the one unity being added at each increment of increase is bounded and indivisible, is irrelevant to the infinity which involves the act of increase. That the increasable amount is bounded, restricted to exclude fractions, is not a limit to the infinity itself. Nor is the fact that a divisible unit is bounded a limit or restriction to divisibility. — Metaphysician Undercover

That's incorrect. Whether or not something is infinite has everything to do with whether or not it is unbounded, because "infinite" is defined as unbounded. Where is your rigorous understanding of infinitiy?. — Metaphysician Undercover

And no, an iterative calculation is not unbounded. It is limited by the physical conditions, and the capacity of the thing performing the iteration. That it is so bounded is the reason why it is not infinite

let arr = [], i;

for(i = 0; arr.length < Infinity; i++){
     arr.push(i);
}

// result: arr = [0,1,2,3,....] (it actually never completes, for obvious reasons)