And if an expert in a field says something then you've reason to think it is true (other things being equal - as usual). — Bartricks

No, that means that you trust him. You trust that he has a justification. On what grounds would you trust him?
We do not trust. We verify.

No, if there are beliefs that cannot be expressed in a language that's irrelevant to whether they will qualify as items of knowledge. — Bartricks

They are formally not knowledge because it will be impossible to verify their justification.

Morality, if it involves any rules, is going to involve normative rules. And is those that require a ruler. — Bartricks

Normative rules lead to rulings, which are simply language expressions. A machine can traverse rules and produce a ruling. There is no need for a human to do that.

I don't know what you're talking about again. — Bartricks

If your conclusion is knowledge, then a machine must be able to reach the same conclusion. Otherwise, it is not knowledge.

You're changing your position. Knowledge involves having a justified true belief, whatever else it involves (actually, I'm sceptical it has to involve that - but it has to involve a true belief). Beliefs are mental states. Machines don't have mental states. Therefore they do not have knowledge. — Bartricks

Beliefs that are not expressed in language or not possible to express in language are ineffable. They are not part of knowledge. You must be able to express the belief, or else it is not knowledge. Hence, legitimate knowledge can always be represented by using language expressions. We do not need undocumented, internal, mental states for that purpose.

So if you're looking up something in philosophy, which is better - Wikipedia or Stanford Encyclopedia? It is Stanford hands-down. Why? It is written by academics - experts in their field. — Bartricks

I have to interject the Church-Turing thesis to your comparison of both sources. Your evaluation is legitimate knowledge only if there exists a purely mechanical procedure to reach that conclusion. Where can we find the description for the epistemically-sound method that you have used to reach that conclusion? We need this in order to supply it to a machine and verify if it reaches the same conclusion as yourself.

I think there are or could be moral rules, but there don't have to be. But rules require a ruler just as values require a valuer. All roads point to a subject. — Bartricks

So, let's forget about any ineffable moralities that cannot be expressed in rules, and limit ourselves to moralities that can. Humans can make inferences based on rules, but are not required for that purpose. Machines are perfectly capable of executing inference engines. Therefore, machines can reach conclusions from a set of rules.

Therefore, I have to reject that "rules require a (human) ruler" or "evaluations require a (human) evaluator":

Rule-based systems. In computer science, a rule-based system is used to store and manipulate knowledge to interpret information in a useful way. It is often used in artificial intelligence applications and research. Normally, the term rule-based system is applied to systems involving human-crafted or curated rule sets. Rule-based systems constructed using automatic rule inference, such as rule-based machine learning, are normally excluded from this system type.

A rule-based system is NOT a human at all, even though humans can do it too (usually with lots of errors, though).

Knowledge requires a belief, machines do not have beliefs, therefore machines do not have knowledge. — Bartricks

Such belief must be expressed in language. Ineffable beliefs cannot possibly be knowledge.
A machine can store language expressions and use them in inference rules.
Hence, a machine can use knowledge.

Like I say, a machine cannot know something because a machine does not have mental states and beliefs are mental states and knowledge essentially involves having them. — Bartricks

Knowledge as a justified (true) belief is a tuple of two language expression: the knowledge claim along with its justification. A machine can perfectly store that. It can also perfectly use it in an inference engine to reach conclusions based on this knowledge.

Knowledge is not restricted to humans. On the contrary, according to the 1936 Church-Turing thesis a problem is only decidable in knowledge if there exists a purely mechanical procedure for doing so.

I do not see how you are challenging premise 2. Does the machine value anything? No, it is a machine. — Bartricks

We would first have to agree that a morality is a set of rules. In that case, derived moral rules are not "valued" but evaluated. A machine is perfectly capable of verifying such evaluation.

From what I gather access to knowledge is restricted and requires a certain combination of dollar digits to unlock. I have nothing against that. People must protect their intellectual property and need money to put food on the table. — TheMadFool

Current technology is driven by open-source software, most of which is freely accessible on platforms such as github.com. If hoarding knowledge were a requirement to put food on the table, then how do you explain that? Are these millions of software developers starving?

In technology, the concept of "intellectual property" is considered to be rather an impediment to progress than to be of any benefit.

Even if wikipedia fails and is inaccurate — TheMadFool

Wikipedia pages are attributed. You can always find the original source in the foot notes. Wikipedia's no original research policy is strictly enforced. That is much more than you can say about the average academic text book.

If you add up the billions of dollars spent on public libraries and university ones around the world, do they produce even a fraction of a fraction of the value that Wikipedia gives away free of charge?

It reminds me of the silly debate in Europe. The governments there want to tax Google, so that they can fund, at great cost, outdated and expensive alternatives to Google search, that cannot possibly compete with Google search, while Google provides its search engine free of charge to its users. They even try to defend that on moral grounds using arguments that invariably end in infinite regress ...

I would personally even pay real money for the privilege of using Google search, while I would never pay a dime for the crap produced by these European governments.

I don't really know what you're talking about now. To know something is to have a true belief about it, whatever else it involves. And computers cannot have beliefs. — Bartricks

A machine can know something, because it can store any knowledge declaration P => Q along with a proof R in first-order logic that proves that Q necessarily follows from P. Therefore, the two-tuple (P=>Q,R) is a justified belief, i.e. knowledge.

So I don't see what problem you're highlighting or why you think expert testimony counts for no more or less than the testimony of an idiot. — Bartricks

If the idiot looks up the tuple (P=>Q,R) in a knowledge database, there is no way in which the expert can refute this tuple. He simply knows nothing more about P=>Q, simply, because there is nothing more to know.

If you believe P=>Q because an expert tells you, then your belief is NOT justified. There is absolutely no reason why it would be knowledge. If you believe P=>Q because an idiot provides you with the two-tuple (P=>Q,R), then your belief is effectively justified. The only legitimate reason why it is justified has been supplied to you, i.e. the tuple (P=>Q,R).

I think moral values are demonstrably subjective. Here is my simple argument:
2. Only a subject can value something — Bartricks

Imagine a morality with three basic rules B={R1, R2, R3}. Imagine that person X uses a statement P in first-order logic that R4 necessarily follows from B. In that case, system S can verify P, and on those grounds accept that R4 is necessarily a consequence of B.

Therefore, system S can objectively decide that R4 is morally required for any person who accepts B={R1, R2, R3}. Note that system S is not even a person. S is just a mechanical device.

Therefore, I have to reject that "Only a subject can value something", because machines are perfectly capable of verifying the first-order logic statement P.

Like I say, if your doctor says the mole is cancerous you are justified in believing it to be cancerous, whereas if your mate Tom says it is cancerous, you are not justified in believing it to be cancerous (even if it is). — Bartricks

The justification for "the mole is cancerous", is never "who" says it, but "how" he says it: justified or not. If the justification for that claim falls within the realm of knowledge, then sooner or later Tom will be a computer system and trivially defeat your doctor. If it is not pure knowledge -- that depends on the precise epistemic context -- then neither an "expert" nor a computer system can systematically be counted on to reach the correct conclusion.

Therefore, this question is not decidable within this context.

There are trivial counterexamples for the type of argument you are making. If Carl is an accountant who believes that the sum is 12535 after adding up a list of numbers, but Tom which is software and which has actually produced the list of numbers, says that it is 11978, then we will believe Tom and not "expert" Carl.

Pure knowledge jobs are shaky things. They were condemned in 1936 already, by the Church-Turing thesis. You must produce something else than the mere application of existing knowledge, in order to stay relevant.

Needless to say, when experts start talking outside their areas of expertise - so, when a neuroscientist starts talking about free will or a biologist starts talking about metaphysics - then you are no more justified in believing what they say than your baker's opinion on these matters. But if you want to know how to bake a loaf of bread, then listen to a baker. If you want to know about what's up with that weird looking mole on your arm, see a medical doctor. If you want to know if you've got free will, consult a philosopher. — Bartricks

Every practitioner of a knowledge discipline faces the threat posed by the Church-Turing thesis:

"A problem belongs to the knowledge domain of beliefs, if there exists a purely mechanical procedure that can reach its solution."

Therefore, if what you do, is based on only knowledge, then what you do, will sooner or later be done by a computer, and you will be replaced by software. The medical profession knows that too. No matter how much political clout they think that they have, we are going to liberally destroy their jobs; and that is a done deal already. Every fake intellectual IYI will have to contend with the same problem.

At the same time, knowledge discovery is protected by Gödel's Incompleteness theorems, by Alan Turing's Halting problem, and by the third millenium problem. So, if your job consists of discovering new knowledge as opposed to merely applying existing knowledge, you will still be relevant in the future.

You see, nothing will stop us from disrupting and destroying, as we please. With cryptocurrencies (i.e. bitcoin) we are kicking out the central banksters, and the commercial fiat banksters, which is another bunch of IYI idiots.

Who can stop us from doing that?

Unless they are protected by the aforementioned limitations, these so-called "experts" are out already. We do not respect merely rehashing existing knowledge, and we shouldn't. These parrots are simply arrogant idiots.

No, there are experts. If your doctor - an expert on the human body and what can go wrong with it - says that the mole on your arm looks dodgy and you should get it checked out, then you're a fool if you think his/her judgement provides you with no better justification for believing it to be dodgy than your mechanic friend's judgement — Bartricks

No, there are no "expert" doctors. There are only justified diagnoses. It does not matter who came up with a diagnose. All that matters, is that it is justified. The only purpose of the entire system of so-called "experts" is to mislead the gullible populace, and to part the fool with his money.

Nassim Nicolas Taleb writes extensively about IYIs, i.e. Intellectuals Yet Idiots, i.e. the academia, the credentialist idiots they produce, and their bullshit ways.

Example: The Intellectual Yet Idiot. What we have been seeing worldwide, from India to the UK to the US, is the rebellion against the inner circle of no-skin-in-the-game policymaking “clerks” and journalists-insiders, that class of paternalistic semi-intellectual experts with some Ivy league, Oxford-Cambridge, or similar label-driven education who are telling the rest of us 1) what to do, 2) what to eat, 3) how to speak, 4) how to think… and 5) who to vote for.

Another example: Charlatans & economists use logical flaw: because a pilot is expert, they are experts. But Pilots are selected via skin-in-the-game mechanisms. Plumbers, dancers, dentists, mathematicians, snipers, pastry chefs are experts. Not this @kaushikcbasu. Economists BS for a living.

Another example: Why do experts, CEOs, politicians, and other apparently highly capable people make such terrible decisions so often? Is because they’re ill-intentioned? Or because, despite appearances, they’re actually stupid? Nassim Nicholas Taleb, philosopher, businessman, perpetual troublemaker, and author of, among other works, the groundbreaking Fooled by Randomness, says it’s neither. It’s because these authorities face the wrong incentives. They are rewarded according to whether they look good to their superiors, not according to whether they are effective. They have no skin in the game.

The entire "expert" concept is bullshit. The USA are bankrupting themselves on exactly that problem in health care. If you believe in that concept, you will simply go bust.

h, I think you've been drinking. Wikipedia is written by people who like pub quizzes, not experts. — Bartricks

There are no experts. There are only justified knowledge claims. Seriously, if it matters who says it, then what he says, cannot possibly matter.

The concept of "expert" is just a ploy to mislead and manipulate the gullible populace. The cognoscenti do not care about who says it. The bitcoin paper was published by an anonymous author. Does it matter? No. What the paper says, is provable from number theory. That is all that matters.

The academia are mostly populated with clowns and circus monkeys, who believe that we should be impressed by their imbecile PhD paperwork and stupid citation carrousels. Sorry, we are not. We only care about justification. We do not care about who says it. We only care about how it is said: justified or not.

I think your beef is with the publishers who make lots of money off peer review publications. And, perhaps, with the disney disciplines who publish each other's work without subjecting it to proper peer review. — Bartricks

No, my beef is with the academia in general, and their obnoxious mentality.

A very typical example of what is totally wrong with their mentality, is the Tanenbaum versus Torvalds debate.

Tanenbaum pointed to his endless list of PhDs and other worthless credentialist dead-tree paperwork to argue that he knew better about operating systems than Torvalds, and that his debile mimix joke of an imbecile piece of crap was better linux.

Tanenbaum did not want to compete on the merits of his work. No, he only wants to compete based on the corrupt citation carousel that he is so proudly engaged in.

But for heaven's sake, whose work powers Android and therefore 80% of the world's mobile phones? Isn't the proof simply in the pudding? Is it linux or his stupid mimix? Seriously, that is the problem with the academia, the vast majority of whom have nothing to show for, besides imbecile paperwork and ridiculous citation carousels. Even though they have nothing to show for, they still know everything better. Seriously, it is so obvious that they know fuck all about operating systems. Zilch. Nada. Nothing.

Criticism from the academia on Wikipedia is of exactly the same nature.

Where is their alternative for knowledge dissemination? They simply don't have one! Knowledge is all about arrows. According to the JtB doctrine, i.e. Justified (true) Belief, it is the following modus ponens that powers it all:

justification statement $\Rightarrow$ knowledge conclusion

Hence, arrows, i.e. links, are the essence of knowledge. Wikipedia supplies us with the ability to link straight to the mainstream knowledge narrative, which in turn, links straight to the original sources that justify it. This is exactly how knowledge is supposed to work.

What alternative do the academia propose? None. They do not have one. Their outdated dead-tree text books and libraries are absolutely unusable in this context. They cannot handle new technology and therefore try to stop its progress. They simply have become an impediment to progress.

The academia sound exactly like the erstwhile middle managers in corporations who complained about the introduction of new technology, couldn't handle it, couldn't work with it, but still insisted that they knew everything better. Corporations, however, tend to smarter than universities. The corporations just fired them all. They terminated their contracts and threw them out of the window. There is no other solution for people who know everything better but objectively have nothing to show for and hence know nothing at all.

Idiots who criticize Wikipedia are more often than not just a bunch of losers who almost surely know nothing, and have nothing to show for, but still insist that they know everything better. They are worthless. Technology just keeps moving ahead while these idiots will sooner or later just get thrown off a cliff.

I'm thinking you've read the proof and worked through it at least some - but maybe not. The universal quantifiers are then qualified via recursion schema. And significantly, while your Prov("X," "S") is recursive, according to Godel, Godel also says the related Provable ("S"); that is, "S is a provable theorem," is not recursive. — tim wood

That is indeed Gödel's original proof, which certainly has its merits, but in the meanwhile various alternatives have emerged, some of which can possibly be considered more straightforward. I specifically like Alan Turing's alternative based on the Halting problem:

Assume that we have a sound (and hence consistent) and complete axiomatization of all true first-order logic statements about natural numbers. Then we can build an algorithm that enumerates all these statements. This means that there is an algorithm N(n) that, given a natural number n, computes a true first-order logic statement about natural numbers, and that for all true statements, there is at least one n such that N(n) yields that statement. Now suppose we want to decide if the algorithm with representation a halts on input i. We know that this statement can be expressed with a first-order logic statement, say H(a, i). Since the axiomatization is complete it follows that either there is an n such that N(n) = H(a, i) or there is an n' such that N(n') = ¬ H(a, i). So if we iterate over all n until we either find H(a, i) or its negation, we will always halt, and furthermore, the answer it gives us will be true (by soundness). This means that this gives us an algorithm to decide the halting problem. Since we know that there cannot be such an algorithm, it follows that the assumption that there is a consistent and complete axiomatization of all true first-order logic statements about natural numbers must be false.

As you undoubtedly noticed, Alan Turing's version does not make use of any particular witness theorem. To some extent, that is a good thing because it spares us from overly focusing on that one witness in Gödel's original proof.

Wikipedia is not peer reviewed and no respectable university will be happy with anyone citing wikipedia in student essays. Wikipedia has its uses, of course - but so do chats down the pub. — Bartricks

The academia are certainly not an absolute reference with regards to knowledge.

On the contrary, their detestable practices concerning locking up publicly-funded research in copyrighted journals make the academia the least appropriate standard for the dissemination of knowledge. We despise their wide-ranging corruption and we view their practices with contempt only.

As I wrote before, only open-source and open-access should be considered as reference material. Seriously, we are justified to spit on the academia and their detestable ways.

As I've often commented, though, I really couldn't care less what the structure of the government is. What I care about is what laws a government does or doesn't have. I'm no more likely to agree with laws just because they're decided by a majority. — Terrapin Station

Totally agreed.

In my experience, countries that are NOT democracies perform much better in that respect. No matter what other stupid laws they usually have on the books, democracies always end up revolving around handouts, freebies, and envy politics.

The best example of a "true democracy" is Venezuela.

The populace started yelling that the price of bread was too high? No problem. Just set the price firmly below the production cost. Now they are complaining that nobody produces bread anymore and that the supermarket shelves are empty.

Of course, Venezuelans still consistently refuse to acknowledge any link between their own political demands and their own misery. It is not their fault at all! Unfortunately, it is not Chavez' fault (or now Maduro's) either because they only did what that the populace of complete idiots demanded from their government, which literally delivered the laws that these retards asked for.

Furthermore, even if the supermarket shelves became stocked again, they wouldn't be able to buy anything there with their worthless Bolivar currency. The Venezuelan government just kept printing money to pay for handouts and freebies, while it is "the rich capitalists" who were running the erstwhile bakeries who should pay for the handouts and freebies !!!

Alexandria Ocasio-Cortez does not only sound like Chavez. She really is like him. You see the same kind of people showing up everywhere in these so-called "democracies". What a joke!

Democracy being moreso bureaucratic than practical makes it freedomness, in my experience. — Shamshir

When does a {hospital} work properly?

When we get the guarantee that the ignorant idiots of the populace won't have a say whatsoever in making {medical} decisions concerning others. Only the cognoscenti will be involved in figuring that out.

When does a {garage} work properly?

When we get the guarantee that the ignorant idiots of the populace won't have a say whatsoever in making decisions concerning {car-repairs on the cars} of others. Only the cognoscenti will be involved in figuring that out.

So, all throughout society we enforce the following principle:

When does {X} work properly?

When we get the guarantee that the ignorant idiots of the populace won't have a say whatsoever in making decisions concerning the {X} of other people. Only the cognoscenti will be involved in figuring that out.

In a so-called "democracy", an exception is made when {X} concerns {X}="how to govern the country". That is a glaring and fundamental mistake. There should be no exception whatsoever, because it is exactly that exception that will start destroying every possible other {X}.

Accordingly, one needs an interpretive layer, such as 'representatives' to examine the will of the people and make as much sense of it as may be, and implement that. — unenlightened

No, sorry, don't even implement that!
It will become a tool for the populace (of mere idiots) to destroy themselves.
Society is not viable like that.

But maybe that's the semblance of the current capitalistic democracy, in which case what would merely democracy be? — Shamshir

In a so-called democracy, the option exists to invent new laws that will make the survival of society itself, impossible. You would think that they would not do a thing like that, but they really do. It is perfectly possible to manipulate the populace into approving self-destructive laws and policies.

Therefore, rule number one:

Never ask the populace (of mere idiots) how to govern the country. Just govern it.

For example, they kicked out the so-called "democratic" government in Thailand and replaced it by a military junta. I can only approve of that. The "democratic" predecessor was busy manipulating the populace into approving principles that would simply destroy that very populace. You would think that the populace would not do a stupid thing like that? Sorry, they always do. Welcome to the real world.

I'm with Churchill - it's the 'least worst option'. — Wayfarer

Pretty much all other options could survive on the long run. So-called democracy can clearly not, as it brings down fertility rates to zero, and eliminates the very society that it is supposed to govern. In that sense, it is not the least worst, but simply the worst option. Only Soviet communism was possibly even worse, but even that is debatable. Furthermore, we will never know, because Soviet communism has imploded already while so-called democracy is still in its very last lap.

Is democracy a tool or a goal unto itself? If it is some sort of goal we reach toward, why should we be reaching for it? — frank

In "Politics and the English language", George Orwell points out that the term "democracy" resists being given a definition. Therefore, it is a member of the class of "meaningless words".

The only purpose of meaningless words is to praise or to black mouth.

"Democracy" is supposedly "good" while "terrorism" is supposedly evil. These terms do not mean anything else but good and evil in the opinion of the person using these words, in reference to an unspecified and undocumented system of fake morality.

It is a conjectural belief that democracy would be a viable political system. I strongly suspect that it isn't.

As far as I am concerned, it leads to the most intrusive kind of government known to mankind, only surpassed by soviet communism in terms of loss of freedom and self-destructiveness.

To be explicit, my main argument being that conceptualization of quantity precedes conceptualization of all maths - in that it is prerequisite to mathematical thought. — javra

In fact, I agree to an important extent with that.

The foundations of mathematics are to some extent considered to be impredicative (circular). However, it is the language of first-order logic that is considered to be the real culprit.

The extending move from propositional to first-order logic is achieved by the introduction of universal quantifiers, such as $\forall$ (the "for each" construct).

What the $\forall$ implies, is the ability to move from element to element in a collection. That looks a lot like assuming a starting point (zero) and a successor function to move away from that starting point; even though all of that actually depends on the unmentioned implementation details of the $\forall$ symbol. Maybe it is possible to practically implement the traversal of a collection without using zero nor the principle of succession? Not sure, and possibly not.

Still, the existence of zero and a successor function are the core axioms in number theory (Dedekind-Peano); which could then already be implicitly pre-assumed in the language of first-order logic. So, number theory could in fact be expressed in a language that implicitly already assumes the existence of number theory, with the problem being hidden away inside the actual implementation of the $\forall$ universal quantifier.

The foundational crisis in (classical) mathematics started raging at the end of the 19th century and was never really solved. The final blows came in 1931 with Gödel's incompleteness theorems and in 1936 with Turing and Church's negative answers to the Hilbert's Entscheidungsproblem.

So, yes, I guess that the foundations of mathematics are likely circular in some way.

Don't - don't, don't, don't - go to Wikipedia for insight. Wikipedia is not an academically respectable source, as your institution should itself have told you. It is shot through with mistakes. Nothing on there is subject to proper peer review. If you are at a university then read from proper peer-reviewed sources - that is, academic articles and books by respectable academic presses (not books by philosopher-wannabes with no academic credentials in the area). — Bartricks

Every phrase in Wikipedia is attributed or attributable.

That is much more than you can say of the average academic textbook, which is just one more way to part a fool and his money.

Furthermore, peer review is just mutual back patting. "I have your back and you have mine." It is very prone to corruption and that is why most peer-review processes are effectively corrupt. Just look at the scandalous academic journals for a good example. They are a corrupt business platform for trading citations. I cite you, you cite my friend, and my friend cites me. It is absolutely trivial to game that system.

In technology, the free and open-source software movement utterly rejects any attempt by the academia to prevent collaboration by re-appropriating copyrights with a view on funnelling them to corporate overlords.

We detest these people, seriously.

Shunning the academia from the knowledge industry is a sheer necessity. Therefore, support Wikipedia.

Only reference open-access research and ignore everything else. The academia are involved in an impossible pile of bullshit with their closed-access journals, and we do not want to have any part in that.

Ignore research published under proprietary corporate copyrights. Join the open-source and open-access movements and reject the academia's self-serving mentality.

I take this as a very 'pure math' position. What do you make of the fact that most math is not pure? I don't just mean science. I mean everyday life. What kind of tip should I leave? How many eggs are left in the fridge? How many more miles can I get with that needle close to E? — joshua

There is nothing wrong with downstream applications, even the most simple ones, but none of that is part of the knowledge discipline of mathematics, which is something different.

I'm not at all against pure-math, just to be clear. But consider the history of calculus. Applications came before rigorous axiomatic theory. Consider also how important intuition is learning math. Even pure mathematicians don't write out complete, formal proofs. They appeal to one another's intuition, and I think most of them aren't formalists at heart. — joshua

If you exclusively deal with mathematics that has a straightforward visual representation, you may indeed develop a more constructivist mentality and even ontology of mathematics. Classical (Greek) geometry tends to be like that. and possibly even number theory. I just pointed out that there are areas in mathematics that are absolutely not like that; especially the disciplines that emerged only over the last century. In my opinion, (real-world) constructivism is even the wrong intuition. Symbol manipulation goes much more to the core of what it is about.

It used to be that people questioned pure mathematics as a merely theoretical exercise with no practical application.

With the spread of mobile phones that is much less the case than before. Almost everybody must have seen source code by now, and realized that it is about annotated abstract symbol streams. It is surprisingly close to the practice of symbol manipulation in first-order logic of mathematics.

Even the old question, Why do I need to spend so much time learning to read and write? is no longer asked by even toddlers who can see all the letters showing up on their tablet and who must wonder what they mean. It has become much more natural for them to learn this. I have never heard my own children ask why they should learn how to read and write.

I think that digital natives do not have an intuition problem with the technology that surrounds them.

No, I'd say those are complex relations regarding quantity. In other words, they would be pointlessly meaningless - correct me if I'm wrong here - in the complete absence of expressions of quantity such as that of "1". — javra

The expression "1" does not appear in set theory or in the lambda calculus (axiomatization of anonymous functions). You can optionally produce the concept of "1" as a necessary result of set theory or of the lambda calculus, but you can happily work in both mathematical theories and derive theorems, without ever mentioning the concept of "1".

Yes, I took calculus in high-school, but I'm no mathematician. Not my thing. — javra

I currently do mathematics as a hobby. For example, if a paper happens to be about Galois theory, I will more often than not read it. Still, I do not need Galois theory for my job. It is just a personal interest. At university, I had to sit, year after year, exams on operational research (linear programming and so on), which is some kind of subdiscipline in mathematics ("optimization"), but I absolutely never used it professionally. I did not dislike it, but I rather treated it like a game of chess. I no longer read anything about optimization because I consider it to be less interesting than other sub-disciplines.

My latest foray in mathematics has been modern Galois theory. It uses a pyramidal hierarchy of terminology.

The Banach-Tarski paradox (BTP) -- possibly a new subject for me -- does that too. To my great frustration, there seems to be very, very little overlap. So, BTP looks like a complete new mountain to overcome. That is why I hesitate so much! ;-)

By the way, it is not because you know one or more mathematical theorems that you understand anything about any other one. It is incredible ...

Mathematics is compelling in computer science, which in turn is compelling in software engineering. So, quite indirectly I deal with mathematics in a professional capacity. With the increasing importance of system security and also cryptocurrencies there has also been a spectacular invasion of mathematical thinking through the theories in cryptography.

Still, I have an exorbitant degree of confidence that none of the above means anything sans representations of unity, aka quantity. A geometric point, for all its marvels of being volumeless, is yet a quantity, for instance. — javra

A lot of modern mathematics is only visible as language, just as a symbol stream about other symbol streams.

For example, I do not see how to create a visual representation for the lambda calculus or the SKI combinator logic. There is nothing, but absolutely nothing visual to it. Seriously, what graphical representation could possibly apply to it?

These mathematical theories are close to computer science, where mathematics tends to be language about language. Of course, there are other sub-disciplines in mathematics that do have a visual representation, but that is not the mathematics that I have dealt with recently.

If the semantics of "quantity" needs better clarification, let me know. Alternatively, if you find I'm mistaken - but understand that "1" represents an idealized perfect integrity, or unity, of existent stuff - please offer some references to maths devoid of notions of quantity (such as the concept of "1", and its derivatives). — javra

The dominant axiomatization in mathematics, ZF set theory (along with AC), does not even mention "1". If you look at its nine axioms, the expression "1" is literally nowhere to be found.

Seriously, run through the symbolic first-order logic sentences for the axioms from first to last and try to find "1" or any other numeral. It is just not there.

There's much more to it than that. We can infer and predict outcomes, based on mathematical analysis of observations — Wayfarer

As soon as you say "observations", i.e. things that you see in the real, physical world, then it is no longer mathematics. In that case, you are doing physics or something similarly real-world.

Whence this predictive relationship between mathematical reasoning and material facts? — Wayfarer

The short story? We don't know.

My own speculation is that the real, physical world is consistent by assumption while mathematics is consistent by construction. So, that may allow for particular isomorphisms between both.

I find Eugene Wigner's essay on it very interesting. — Wayfarer

I think I can agree in globo with what Eugene Wigner writes. For example:

It is, as Schrodinger has remarked, a miracle that in spite of the baffling complexity of the world, certain regularities in the events could be discovered. The preceding discussion is intended to remind us, first, that it is not at all natural that "laws of nature" exist, much less that man is able to discover them. The reason that such a situation is conceivable is that, fundamentally, we do not know why our theories work so well. Hence, their accuracy may not prove their truth and consistency. The reason that such a situation is conceivable is that, fundamentally, we do not know why our theories work so well.

Agreed. Science does not describe the unknown construction logic of the real, physical world, i.e "the true laws of nature". Science rather describes observable patterns that are incredibly resilient to experimental testing. These two things are obviously not the same. We are talking about science as a really useful and clever hack. Science will indeed probably never manage to become the theory of everything (ToE).

Every empirical law has the disquieting quality that one does not know its limitations.

Agreed. A method that observes the real, physical world is itself part of an abstract, Platonic world, and can therefore never observe itself. This is indeed the main weakness of empiricism.

Mathematics, or, rather, applied mathematics, is not so much the master of the situation in this function: it is merely serving as a tool. It is true, of course, that physics chooses certain mathematical concepts for the formulation of the laws of nature, and surely only a fraction of all mathematical concepts is used in physics. However, it is important to point out that the mathematical formulation of the physicist's often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena. This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language.

Agreed. If the real, physical world is consistent -- on falsificationist grounds it is perfectly sound to assume this -- then the consistency-maintaining bureaucracy supplied by mathematics must indeed be the correct language.

I do object to Brouwer's direct, constructivist connection between mathematics and the real, physical world. Unlike physics, mathematics does not deal with the semantics of the real, physical world, but only with the consistency of abstract, language expressions. Furthermore, if you can test it, you should test it. So, mere symbol manipulation is not the main instrument for the analysis of the real, physical world.

The reader may be interested, in this connection, in Hilbert's rather testy remarks about intuitionism which "seeks to break up and to disfigure mathematics,"

Disagree. Hilbert does not disfigure mathematics. He rather seeks to maintain its purity. The real, physical world or any connection to it, is not a legitimate subject in mathematics. People who are interested in the real, physical world should rather seek answers in physics or other scientific disciplines, because mathematics is not about that.

I propose to refer to the observation which these examples illustrate as the empirical law of epistemology. Together with the laws of invariance of physical theories, it is an indispensable foundation of these theories. Without the laws of invariance the physical theories could have been given no foundation of fact; if the empirical law of epistemology were not correct, we would lack the encouragement and reassurance which are emotional necessities, without which the "laws of nature" could not have been successfully explored. It is therefore surprising how readily the wonderful gift contained in the empirical law of epistemology was taken for granted.

Agreed. Science is a knowledge-justification method. Hence, epistemology really matters to science. It is obviously unavoidable.

Still, epistemology is only "empirical" about the abstract, Platonic world of knowledge, and not about the real, physical world. So, the use of the term "empirical" is a bit ambiguous in this context. Unfortunately, there does not seem to be a proper term for what epistemology does. The term that comes closest, is indeed "empirical", but it requires making a perspective shift from "real world" to "world of knowledge".

All the same, in a broader sense of language, how is mathematics - which is codified quantity and relations between quantity (right?) — javra

No, I disagree because most of mathematics is not about codified quantity or relations between quantity. Only number theory to some extent is. Furthermore, the dominant axiomatization is currently set theory, and that has been the case for almost a century.

The ontology of mathematics is not a settled matter. Everybody still disagrees with everybody else. Still, the idea that it would be about quantity is considered very, very restrictive. For example, is a combinator or a function related to quantity? In another example, I don't think that, for example, category theory even ever mentions quantities. It is rather about structures, mappings between these structures, and possible preservation of structure. I don't think you'd ever see a quantity in that context.

not an abstracted form of language employed by humans for various purposes? — javra

It is language and permissible manipulation of the symbols in the language. However, that does not settle the matter of the ontology. Far from.

The ontology of mathematics is an outstanding problem.

His disagreement was with the notion of 'observer dependence' which physics suggested. He was never reconciled to that. — Wayfarer

The analogy with random numbers is exactly that: observer dependence. The randomness of numbers depends on the observer. They are not (necessarily) intrinsically random (which is something we do not even know). That is where Kant's epistemic consideration kicks in:

Prolegomena, § 32. And we indeed, rightly considering objects of sense as mere appearances, confess thereby that they are based upon a thing in itself, though we know not this thing as it is in itself, but only know its appearances, viz., the way in which our senses are affected by this unknown something.

Therefore, if something consistently appears to you in a particular way, then you can proceed with the idea that they are in that particular way. What else are you going to do anyway? So, I side with Bohr.

1 + 1 = 2 is not pure mathematics. It is a fact that is thoroughly entwined with the reality in which we live. — javra

These are empirical patterns in which people detect some form of consistency. Mathematics is only about that consistency, and nothing else. It is not empirical. The language expression "1+1=2" is handled by math, because it is language. What you see in the real, physical world, is not handled by math.

According to his learning, one can easily construct a coherent theoretical mathematics that blatantly contradicts everyday aspects of reality such as that of gravity. Axioms are what you want them to be and you simply construct from them. — javra

Of course. Language can create abstract, Platonic worlds that have nothing to see with the real, physical world. That is not even specific to mathematics. Science fiction movies do that too. The difference is that mathematics has an extremely elaborate notion of consistency. Mathematics will still not allow you to contradict yourself inside that world without gravity. Otherwise, anything flies, because that is the name of the game.

Note the reflexive equivocation of 'real' and 'physical'. You write from a perspective which assumes the reality of the sensory domain - as us denizens of a sensate culture are inclined to do! But what if the source, and the goal, of metaphysics is not within that domain at all? — Wayfarer

Well, we can discuss Matrix-like philosophies, but they are not knowledge, because they are about the real, physical world but not justified from anything else therein.

I am a fan of the Matrix idea. I like it quite a bit, but I also know the limitations of that kind of thought exercises, and I do not classify them as knowledge or meta-knowledge. These things are hypothetical conjectures. Si non vere, bene trovato.

You keep repeating this, as a dogma, and oppose anyone who challenge it as 'constructivist heretics'. — Wayfarer

Well, in a sense, I am just continuing the Brouwer-Hilbert controversy, but in the end, there is rather a consensus that Hilbert won that debate. So, yes, I believe that Hilbert was right and that Brouwer was satanically wrong.

But the weakness of this claim is that we're utterly surrounded by devices and technologies which would not exist, were mathematics not applied to the physical world. — Wayfarer

Yes, but it always goes through a downstream discipline which makes the necessary perspective shift.

For example, mathematics is distinct from computer science (CS), because CS has to contend with equipment that does not compute instantaneously fast nor has infinite memory. Mathematics does not have to do that. Engineering also has to deal with real-world trade-offs and cost considerations. There you have again other elements that kick in. It is the purity itself of mathematics that creates the need for downstream disciplines that are not completely pure.

The same overall principle that forces sanity in mathematics: accord to our experiences of what is. No? — javra

Mathematics has nothing to do with real-world experience. It is completely divorced from it. It is about consistency in abstract language-only expressions. Seriously, if a claim is about the real, physical world, then it is not mathematics.

Theoretical (pure?) mathematics can get a little disjointed from reality at times, last I heard. — javra

There is no other mathematics left than pure mathematics. If it is about reality, then it is a downstream discipline, such as physics or another subdivision in science, or engineering, or some other downstream activity.

Mathematics does not compete with physics or with science in general. That is epistemically impossible.

As to metaphysics, as an abstract principle to be ideally pursued, make its affirmations falsifiable via reasoning and/or experience. — javra

In that case, it tries to compete with disciplines that do that already; and it miserably fails.

And I am of the view that philosophy requires no apparatus. — Wayfarer

I agree for epistemology, even though it still needs access to the body of existing knowledge in order to detect patterns in it; which Immanuel Kant more than successfully did. Karl Popper also did that magisterially.

Pointing out an existing meta-knowledge pattern in knowledge is important, because it allows the downstream knowledge practitioners to take note of these findings in their quest for the discovery of new knowledge.

That is the difference between epistemology and metaphysics. Unlike epistemology, there are no downstream practitioners who need any output from metaphysics. It just gets ignored, and that has been the case for almost 2500 years now.

On what merits to you deny that the study of reality is a metaphysical issue? — javra

There is no knowledge possible without an epistemic knowledge-justification method; two of which, in the context of the real, physical world, are science and history. So, there we have two epistemically sound knowledge domains.

Epistemology, which does not study the real, physical world, but the abstract, Platonic world of knowledge is also sound, because it faces an already committed world of knowledge as it has emerged already. Hence, the detection of patterns in these existing commitments is not merely spurious.

What principle would force a bit of sanity in metaphysics?

In my opinion, there are no constraining structures that enforce soundness in the activity of merely investigating presuppositions; not even the effect on theorems that already make use of these presuppositions.

It is possible to investigate presuppositions in mathematics, because they are freely chosen or even arbitrary, and they do not claim to be the construction logic of the real, physical world anyway. A good example is the analysis on why the axiom of determinacy is incompatible with the axiom of choice in set theory.

The construction logic of the real, physical world, i.e. the theory of everything (ToE) is unknown. Therefore, a similar exercise is impossible in that context. Still, that is what metaphysics tries to do in vain.

He doesn’t say that, well at least not in the quote you provided. He said ‘knowledge frameworks’ and I think it’s a perfectly valid point. The Bohr-Einstein debates were basically metaphysical in nature, and Bohr’s ‘Copenhagen interpretation’ is arguably Kantian in many respects. — Wayfarer

Kant never tested anything experimentally. Therefore, his views cannot be classified as physics in its modern understanding, which already prevailed in the 1930ies, when Collingwood mentioned the term "Kantian physics".

Concerning the Bohr-Einstein controversy, if you receive a sequence of numbers that look random, even though they may have been generated by a pseudo-random number generator, which uses a seed to construct the sequence, and the knowledge of which would allow you to flawlessly predict the next pseudo-random number, then in absence of knowledge of such seed, I recommend to treat these numbers as random. Therefore, I side with Bohr. Furthermore, I do not consider this question to be metaphysical at all.

My issue lies within the dichotomy: you either have to lie, or you have to tell the murderer where your children are. — Clint Ryan

No matter how much I like Kant's "Critique of Pure Reason" for its ability to detect interesting epistemic patterns in the world of knowledge, I find his publications on ethics, such as "Critique of Practical Reason" to be much, much weaker. It doesn't seem to correspond particularly much with how morality is done in practice.

Furthermore, I personally relegate ethics and morality to the axiomatic domain of religious law, because it has a much more elaborate infrastructure for determining jurisprudential questions, i.e. the "What is right and what is wrong?" type of questions on human behaviour. In that sense, I find Kant's work on ethics also a bit irrelevant to me.

In this particular case, Kant simply fails to take the doctrine of the lesser evil into consideration.

For example, most activity in medicine revolves around administering poisons and liberally cutting into people's bodies. Therefore, absolute rules such as "You shall not administer poisons" or "You shall not chop off other people's limbs" are nonsensical. In such case, you could as well close all hospitals, because that is pretty much all they do.

When faced with a dichotomy between two evils, you simply choose the lesser one.

So, yes, there is an interdict on bearing false witness, but it will trivially take a back seat on avoiding to do something worse than that. If Kant's views on ethics cannot handle that in all obviousness, then they are simply unworkable.