I would point to capital, and the drive of the wealthy to increase their share of capital, on the backs of workers. — rlclauer

In my opinion, we are long past that 19th-century, early 20th-century conflict between capital and labour. Factory work has either been automated or shipped off to China and the like.

I understand you are critical of the state and the type of education on offer at the moment, but do you agree with this one point? To put it even simpler: good conditions generally improve outcomes and vice versa. In my opinion that is not controversial. In fact there is a mountain of evidence to support that claim. — rlclauer

The middle class eagerly adopted "the system", and being born in the middle class made you likely to do relatively well in life, in a sense that you would probably end up middle class too. Still, that would have been the case also without "the system". Rich people play golf. So, playing golf will make you rich ... not really.

To an important extent, the middle class is gone now anyway. So, why waste time on something that is dead already?

Nowadays, if your dad is a plumber, and he takes you with him as an apprentice, you will almost surely do better than if you went to college. By the way, that is how it used to work, before they introduced "the system". For the vast majority of people, it would work better if they just reintroduced it. Furthermore, the need to put children in large holding pens and other large-scale nurseries, because the parents could not take them with them to the factory, is also mostly gone anyway.

With the boys incessantly beta-orbiting their unattainable targets, i.e. girls who are not interested in effeminate soy boys who are addicted to Ritalin -- since the age of six -- meant to cure their imaginary ADHD problem, we can also say goodbye to the nuclear family, I think.

Every day, the mess keeps growing worse. For example, for someone who is gender-confused, it is really hard to make headway on any nuclear-family ambitions. Is that person supposed to become the husband or the wife? If this person cannot answer that question, then what is the other side supposed to do?

f your "poor victims" didn't preach indoctrinate proselytize mutually inconsistent superstitions day in and day out, then there wouldn't be a whole lot of Hitchens'ses around to disabuse those postulates. — jorndoe

Hitchens is like someone who goes through life claiming that he does not like to play or watch tennis. Fine, I should say: then play or watch cricket, or whatever. But no, that is not what Hitchens wants, because he declares himself to be a believer in non-tennis. So, what are tennis aficionados supposed to say to Mr. "non-tennis" Hitchens, besides: "Why don't you get a life?"

I view humans as acting out their behavior which was determined since childhood, which is why I emphasized education as a means to assist people in realizing their potential. — rlclauer

Well, it is certainly not the State-controlled education system that would teach them how to use Google maps. In fact, if you look at it, trillions of dollars later, what exactly have their students/customers learned? You see, these State-controlled schools were designed halfway the 19th century and haven't changed since. Do you know of anything else that managed to avoid changing for over a century? With every year that passes, the gap and the disconnect become even worse. They are sending their graduates straight to the unemployment queue.

I increasingly see the State-controlled education system as detrimental to society. It has become a tool to destroy people's potential.

They put boys and girls together in the same class. To cut a long story short, it turns the boys into idiots and the girls into sluts. By making them unsuitable for any kind of long-term relationships, the State-controlled education system is single-handedly destroying sexual reproduction across society, with marriage and birth rates collapsing. They indoctrinate the boys to be like girls, and the girls to be like boys. The effects are disastrous. They are turning the next generation into very, very unhappy people.

Metaphysicians do not use any specific system, it is more like intuition, so metaphysics appears to be random nonsense to the uninitiated. — Metaphysician Undercover

I am heavily "epistemized" and deeply invested in the idea of the existence of various knowledge-justification methods. Without such method, it is not knowledge.

Still, I completely acknowledge that non-knowledge mental faculties are key, not just for the discovery of new knowledge, but in general. But then again, systematization means converting things into knowledge. If it is not knowledge, but rather intuition, this is guaranteed to be a failing strategy.

f, for simplicity sake, we generalize and call this intuition, then we have something named, which we can discuss, and analyze toward understanding it. — Metaphysician Undercover

There cannot be knowledge, i.e. a justified (true) belief, about intuition, because in that case it would be knowledge and not intuition.

We can say now, that principles, axioms, are not chosen arbitrarily, but they are chosen by intuition. Intuition would assess the applicability of various possible principles, in relation to various goals, ends. — Metaphysician Undercover

I have run into at least two research fields where the goal was to redo a particular axiomatic system with arbitrarily-chosen subsets of its axioms.

Hilbert calculi are like that. You cripple first-order logic by removing some of its construction logic, and then you check what's left. It is very interesting. The point is to show that it is perfectly legitimate to leave out whatever you want, and go with the remainder, and see where you get.

Second-order arithmetic (Z2) is a similar research topic. Cripple arithmetic by adding/removing rules, operators, and so on, and see where you get

In the end, this kind of research rather amounts to playing with "cool toys". But then again, it is not possible to know what people will find unless they actually try. Furthermore, this type of research nicely emphasizes the true nature of axioms as fundamentally arbitrary starting points.

Anything more on offer more central to the OP, the philosophy of Software Engineering? :chin: — Pattern-chaser

The philosophy of software engineering? Oh boy, that sounds exciting. Where do I sign up? — S

I have had similar heated discussions, mutatis mutandis, with programmers who get up in arms because I refuse to learn AngularJS and/or ReactJS, because I prefer to stay on JQuery, hoping that the current fads will be replaced by new fads before I will have been forced to waste time on them. Same story for MongoDB. The NoSQL hype seems to have subsided. I have superbly managed to avoid it. Congratulations to myself.

I did do a stint in Docker, and I actually regret it now, because you don't really need Docker to do program closures, because you can do all of that with much simpler tools. Furthermore, Docker is just being abused as a glorified installer anyway.

So, I hereby retract every positive thing that I ever may have said in the past about Docker. So, what else have I successfully managed to avoid? A lot of stuff, actually. Too much to mention ... I am so proud that I do NOT know these things. I just don't. Am I great, or what? ;-)

In partial agreement with you but there's one area of philosophical argumentation that Hitchen's Razor is extremely useful viz. the issue with burden of proof. I'm familiar with it from the God debate (theism/atheism). — TheMadFool

In terms of epistemology, i.e. when defining knowledge as a justified (true) belief, the appropriate procedure is a bit less trivial than just asking for "burden of proof". If we define an epistemic domain as the collection of knowledge claims that can be investigated using a particular knowledge-justification method, then the procedure goes as follows:

[1] What epistemic domain does the question actually belong to, the main epistemic domains being axiomatic, scientific-falsificationist, and historical ?

[2] Is the question within reach of its knowledge-justification method?

[3] Ok, if yes, only now try to solve the question by using the knowledge-justification method that applies to it.

With [1] the biggest problem is that most people, obviously including Hitchens, are simply not aware of the existence of different non-overlapping epistemic domains. They seem to assume that there is only one epistemic domain, namely, science.

Scientism is an ideology that promotes science as the only objective means by which society should determine normative and epistemological values. The term scientism is generally used critically, pointing to the cosmetic application of science in unwarranted situations not amenable to application of the scientific method or similar scientific standards.

People like Hitchens, who clearly suffer from scientism, think that they are smart, while they are incredibly stupid. That is the result of the Dunning-Kruger effect:

In the field of psychology, the Dunning–Kruger effect is a cognitive bias in which people mistakenly assess their cognitive ability as greater than it is. It is related to the cognitive bias of illusory superiority and comes from the inability of people to recognize their lack of ability. Without the self-awareness of metacognition, people cannot objectively evaluate their competence or incompetence.

Of course, people like Hitchens would not dare to misplace the axiom of infinity in the scientifc-falsificationist domain, and insist on a simplistic "burden of proof" for it, but they are definitely bold enough to do that with religion.

Step [2] is whether the question is actually decidable by its knowledge-justification method. In the 1930ies, enormous progress was made in the realm of decidability, simply because David Hilbert asked the entire field of mathematics to work on his Entscheidungsproblem, i.e. the Decidability Problem. It is the combination of Kurt Gödel's, Alan Turing's, and Alonzo Church's work that yields the Church-Turing thesis. As you can imagine, most questions are actually not decidable. So, nowadays we have that enormous field of computability (=decidability) the existence of which people like Hitchens are not even aware.

Hitchens regularly said things that already in the 19th century were considered to be stupid. He was so ignorant.

Concerning [3], even if the question is decidable, it could still take 350 years to finally solve Fermat's Last Theorem. Hitchens, on the other hand, always had an answer ready within ten seconds. In his mind, difficult questions did not exist, because he was always sure that he knew the answer, even when he didn't.

I do not have a dog in this fight, but it seems like Quod grātīs asseritur, grātīs negātur is valid. I can claim there is intelligent life on 23 planets, but I make this claim without evidence. There are quite a few planets that MIGHT POSSIBLY host life of some sort, and there is evidence for that claim. But there is no evidence at all for the claim that 23 planets host intelligent life. So you can say, "No there are not." — Bitter Crank

A knowledge claim is an arrow between a starting point and a conclusion. In the empirical realm -- life on other planets is clearly a claim about the physical world -- it is an arrow between observations and a conclusion that follows from these observations.

In an axiomatic domain, a knowledge claim is an arrow between a starting-point rule and a consequential rule. The starting point simply does not consist of observations but of a rule. For example, if there is a countable infinite number of natural numbers, ∞, then the number of real numbers is 2^∞.

Hitchens was using his so-called razor, not to argue that the consequential rule does not necessarily follow from the starting-point rule, but to attack the starting-point rule itself. In the example above: There is absolutely no reason to believe that there is an infinite number of natural numbers, i.e. ∞. This is obviously true, but that is not what it is about.

Attacking the knowledge claim "if cardinalityOf(N) is ∞ then cardinalityOf(R) is 2^∞" cannot be achieved merely by rejecting "cardinalityOf(N) is ∞", and also not by rejecting "cardinalityOf(R) is 2^∞". You have to conclusively show that "cardinalityOf(R) is 2^∞" does not follow from "cardinalityOf(N) is ∞".

So, if Hitchens' so-called razor makes sense, you can use it to successfully attack the axiom of infinity, or any axiom in ZFC, because none of ZFC's axioms, i.e. the foundations of axiomatic set theory, can be justified with any evidence.

Of course, Hitchens did not dare to attack mathematics on those grounds. He pointed his arrows to a seemingly easier target: religion, but it is obviously the same attack. Hitchens' views are epistemically unsound, and clearly invalid, but obviously still popular with other atheists, who will defend them, because they like his conclusions. As I already pointed out, Hitchens was a cherished accomplice of Satan.

I am suggesting while that yes there is definitely uniqueness to each person, if given the right education, diet, and living in a society which is safe and contains opportunity for them, they will be dramatically different, and for the better. — rlclauer

Well, a few blocks away from own place, they rent out dwellings at $60/month. Furthermore, people who eat for less than $1/day tend to be much healthier than myself. Not eating meat more often than a few times per month is a good thing, and fasting once in a while for a week or so, is even better. So, a complete family can survive here for less than $100/month.

I do not say that one should necessarily strive to be poor, but people seriously exaggerate their problems.

Concerning opportunity, I made all my money from doing things on the internet.

So, if you want to replicate that, you need to learn basic reading/writing skills, preferably at a young age, and figure out how to Google search as to browse for things, and then you can make things snowball from there. By the way, you are not going to be better at programming than a shantytown kid who learned it from playing with his $150 mobile phone and $5/month internet connection. Even the poorest kid over here, is playing with a mobile phone before he can even walk.

So, if it is that easy, why don't they do it?

Taxi and tuktuk drivers stubbornly and systematically refused to learn how to use Google Maps, no matter how many times customers may have pointed out the issue to them, until the Grab ride-sharing network started operating here. Suddenly, all of them can now use it. Seriously, each one of them. No exceptions. You do not even need to ask them to use it. They are already using it just to find you in order to pick you up. So, the Google Maps war is finally over now. I lost every Google Maps battle in the last ten years, but apparently I still won the war.

The locals here consider me to be some kind of magician. What I do, in their eyes, only works because it is me doing it. If they tried it, it would not work. That is the gist of their opinion. I don't have a problem with that, because I don't feel the need to interfere with what other people do. I do my thing and they do theirs. Furthermore, I should carry a magic wand around, because they will probably think that I know the secret of how to turn people into frogs.

Which is why this razor wouldn’t affect it — khaled

Yes, of course. People like Hitchens are obviously no credible threat to the field of mathematics, if only, because they wouldn't survive for thirty seconds if they had to steer their own ship through uncharted waters on the high seas. It is just that I do not like people like Hitchens, whose only goal in life is to discredit and otherwise viciously attack other people. Hitchens was a cherished accomplice of Satan. Richard Stallman said about Steve Jobs: "I am not glad that he is dead but I am glad that he is gone." About Hitchens, I rather abbreviate all of that to "dead and gone", and we wouldn't want it any other way.

It gets used broadly now in philosophical discussions. So even if the original was aimed at one issue, it is used in general. — Coben

Well, ever since the annexation and reappropriation of logic by mathematics, the remaining flagship sailing for the colours of philosophy is epistemology.

If knowledge is defined as a justified (true) belief, then knowledge has the shape of an arrow. Therefore, we do not reject a knowledge claim because we do not like its starting point. We also do not reject a knowledge claim because we do not like its conclusion. We only reject it because the conclusion does not necessarily follow from the starting point.

Therefore, Hitchens' approach in which he arbitrarily rejects starting points, is just a cheap slogan that he could use and abuse to reject pretty much any knowledge claim. The late, dead Hitchens was a rhetorical attack dog, with a strong emphasis on the word "dog". May his carcass rot in hell.

So you're off your reservation with the wrong opinions on the wrong topics on and about which you don't have adequate information, knowledge, or understanding. And predictably, you're thereby dismissive and defensive - very weak stances from the standpoint of rhetoric. Of course from your area, it's simpler: you're just plain wrong. — tim wood

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Exsurge Domine
Condemning the Errors of Al Contali
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We can scarcely express, from distress and grief of mind, what has reached our ears for some time by the report of reliable men and general rumor; alas, we have even seen with our eyes and read the many diverse errors. Al Contali's errors are either heretical, false, scandalous, or offensive to pious ears, as seductive of simple minds, originating with false exponents of the faith who in their proud curiosity yearn for the world’s glory. We can under no circumstances tolerate or overlook any longer the pernicious poison of the above errors without disgrace. No one of sound mind is ignorant how destructive, pernicious, scandalous, and seductive to pious and simple minds these various errors are. Therefore we, in this above enumeration, important as it is, wish to proceed with great care as is proper, and to cut off the advance of this plague and cancerous disease so it will not spread any further. With mature deliberation on each and every one of Al Contali's theses, we condemn, reprobate, and reject completely each of these theses or errors as either heretical, scandalous, false, offensive to pious ears or seductive of simple minds!
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In my opinion, much of these outcomes are the products of the structure of the society, and other factors which have nothing to do with someone's merit or lack thereof. — rlclauer

In my own case, I do not feel that I have particularly been obstructed or favoured in SE Asia, while doing the things I have been doing. It's not that I can turn around and point to some evil society that threw a spanner in the works. It has worked out absolutely fine.

Of course people prefer to live in comfort rather than a pile of garbage, but the whole subject I am trying to get at, is what produces these outcomes? — rlclauer

On the short run, outcomes look arbitrary and heavily influenced by the environment. On the long run, however, there is probably a real pattern to it. The common denominator is probably yourself.

My own take is as following. If you ask a teenage girl why she is wearing these clothes, she will say: because all my friends are wearing them too. I think that this pattern is actually very general. Failure-inducing behaviour gets often copied wholesale from others. Furthermore, in the background, there certainly are people who benefit from spreading lies and manipulating the masses. I have always been quite immune to manipulative mainstream media and education systems, but I do not really know why.

Hence, I believe changing the way the economy is structured from the ground up can generate more equality from the starting point, which will translate to more equality at the finishing line, and I believe considering how individuals are constrained or propelled by their circumstances is a good way to see that the current system generates illegitimate hierarchies. — rlclauer

Wherever there is a herd of sheep, you will see packs of wolves materializing out the blue. Wherever there are gullible people, you will see manipulators gearing up to manipulate. Go to facebook and watch how the adverts, commercial, social, and political start flying in your face. Pick https://www.reddit.com/r/popular. It won't take long before you will read or see very manipulative messages. You won't see them at https://www.reddit.com/r/epistemology because the crowd there is less manipulable. So, the manipulators avoid wasting time there, because their attempts would be pointless anyway.

You will not be able to change anything to that phenomenon. Even if you change the hierarchy, the same crowd is going to sink to the bottom, and the same crowd is going to rise to the top. The wealthy datsha bureaucracy of the Soviet Union were obviously the former factory owners, while the factory workers themselves became even worse off than before.

Fair enough, then. You know nothing whatever about rhetoric or its subjects. Your subjects are all apodeictic — tim wood

Well, yeah, probably. So?

Agreed, Hitchens's razor is a pig in the parlor of mathematics, but in rhetoric a fine and useful tool. And in rhetoric, your "axioms" (quotes because yours is a term of art) non-sequiturs. — tim wood

Well, a good part of the body of modern knowledge is actually quite counter-intuitive, when you think of it. That is undoubtedly why Hitchens, who is completely ignorant of its caveats, sounds so ignorant. Hitchens was someone who took great pleasure in depicting other people as idiots, but his own views were clearly even worse.

When there is an accumulation of gross wealth at the top, deaths of despair and, for the first time in the history of developed countries, decreasing life spans in many regions, and the rise of populism, I think to say it is working just fine is highly inaccurate. — rlclauer

Well, I did not say that the 100+ national systems, that work fine in my opinion, are the ones of developed countries. I certainly do not live in one. I tend to live in SE Asia. I have been here for longer than a decade now. For various reasons, I do not like living in developed countries. My own experience of living in the European Union is quite negative.

Unless what you mean by "working just fine" is, "well me and the people I care about still got our paychecks," then if that's what you mean, sure, I could see why you would think that way. — rlclauer

Of course, I survived until now. So, I have clearly managed to make and spend enough to stay afloat.

I personally think that they have got it all wrong in "developed" countries. They are not even that much more "developed" any longer. The gap is gone, really.

It is rather that each country has segments of the population that move at different speeds. Some people here still live as subsistence farmers, while others fully participate in the global economy; pretty much undistinguishable from what people do in New York or London.

Then the question rather becomes: What do you prefer? Bangkok or New York? Well, in my case, Bangkok any day of the week. Furthermore, anybody I meet here in SE Asia who has also previously lived in a developed country (EU, USA, ...) does not want to go back. You would have to drag them back by their hair, kicking and screaming.

This is Procrustean - and a variety of category error. Your "axiom" is clearly a term of art, properly restricted to its limited area. Which "area" has nothing whatever to do with Hitchens's razor or its applicability. Perhaps you've slipped on the various distinctions to be made in the meaning and usage of the word "axiom." And is my assumption about your understanding of rhetoric reasonable? It appears not to be. — tim wood

Hitchens' razor just expresses that he does not understand mathematics, science, engineering, nor the link between these domains. By design, it all starts from arbitrary rules with zero justification. Hitchens simply had no clue about the true nature of modern knowledge.

You see, the flagship of mathematics is, beyond any doubt, general abstract nonsense, i.e. category theory. It has absolutely arbitrary starting points (axioms), and, to the non-mathematician, leading to pretty much absurd conclusions.

These things are not something for people like Hitchens. That is why he produced that kind of low-knowledge "razor".

All the razor is doing is saying: sure you can start with this axiom, or this one, or that one, as long as there’s no evidence for them they’re all equally worthless. — khaled

Mathematics does not make any claim as to usefulness or meaningfulness. That is so by design.

Here's a definition: "Axiom definition, a self-evident truth that requires no proof." — tim wood

That is a wrong definition outdated now for almost a century. The following is the long story of the role of axioms in mathematics and what they mean:

A philosophical defeat in the quest for "truth" in the choice of axioms

Hilbert's axiomatic system – his formalism – is different. At the outset it declares its axioms.[15] But he doesn't require the selection of these axioms to be based upon either "common sense", a priori knowledge (intuitively derived understanding or awareness, innate knowledge seen as "truth without requiring any proof from experience"[16] ), or observational experience (empirical data). Rather, the mathematician in the same manner as the theoretical physicist[17][18] is free to adopt any (arbitrary, abstract) collection of axioms that they so choose. Indeed, Weyl asserts that Hilbert had "formaliz[ed] it [classical mathematics], thus transforming it in principle from a system of intuitive results into a game with formulas that proceeds according to fixed rules".[19] So, Weyl asks, what might guide the choice of these rules? "What impels us to take as a basis precisely the particular axiom system developed by Hilbert?".[19] Weyl offers up "consistency is indeed a necessary but not sufficient condition" but he cannot answer more completely except to note that Hilbert's "construction" is "arbitrary and bold".[19] Finally he notes, in italics, that the philosophical result of Hilbert's "construction" will be the following: "If Hilbert's view prevails over intuitionism, as appears to be the case, then I see in this a decisive defeat of the philosophical attitude of pure phenomenology, which thus proves to be insufficient for the understanding of creative science even in the area of cognition that is most primal and most readily open to evidence – mathematics."[19] In other words: the role of innate feelings and tendencies (intuition) and observational experience (empiricism) in the choice of axioms will be removed except in the global sense – the "construction" had better work when put to the test: "only the theoretical system as a whole ... can be confronted with experience".

In other words, axioms have fundamentally been arbitrary rules since the first half of the 20th century. They are certainly not correspondence-theory "true" in any way.

Have you perhaps slipped on the distinction between evidence of and proof of? — tim wood

Axioms are by definition asserted without evidence.

The English version is: What is asserted without evidence can be dismissed without evidence and is attributed to the famous atheist late Christopher Eric Hitchens (13 April 1949 – 15 December 2011). — TheMadFool

In that case, axioms are not legitimate. If axioms are not legitimate, then mathematics is not legitimate, because mathematics is exclusively axiomatic. If mathematics is not legitimate, we must remove the entire bureaucracy of formalisms that maintains consistency in science and engineering. In that case, science and engineering will mostly go out of the window.

In the meanwhile, we are already back in the stone age.

Informal mathematics means any informal mathematical practices, as used in everyday life, or by aboriginal or ancient peoples, without historical or geographical limitation. Modern mathematics, exceptionally from that point of view, emphasizes formal and strict proofs of all statements from given axioms. This can usefully be called therefore formal mathematics. Informal practices are usually understood intuitively and justified with examples—there are no axioms. This is of direct interest in anthropology and psychology: it casts light on the perceptions and agreements of other cultures. It is also of interest in developmental psychology as it reflects a naïve understanding of the relationships between numbers and things.

In other words, Hitchen's razor is something suitable for aboriginal tribes in the one or the other virgin rainforest only.

"Irrational" refers to an incommensurable ratio. This means that the two things being related to each other cannot be measured by the same system of measurement, such as the examples I gave you, the circumference and diameter of a circle, as well as the sides of a square and it's hypotenuse. What this indicates is that there is incommensurability between one spatial dimension and another. — Metaphysician Undercover

Well, the link with classical, Euclidean geometry has long ago been abandoned in contemporary number theory. I suspect that it was completely gone by the end of the 19th century, at the same time as they dumped Euclid's Elements. I have never had to carry out arithmetic using a straightedge and compass, like the Greek in antiquity apparently did.

You mean the problem can be solved by hiding the infinite regress behind "complicated operations". A good metaphysician is trained to recognize such sophistry. — Metaphysician Undercover

Well, taking a square root is no longer basic arithmetic, and therefore considered "more complicated". It is not that even a simple calculator cannot do it. These operations got historically, gradually introduced in order to solve problems. Actually, Pythagoras already needed square roots.

We need some intuition as to which of the proposable principles are credible. — Metaphysician Undercover

Yes, I did refer to non-knowledge mental faculties. Intuition is clearly one.

This is where new knowledge comes from, determining errors in the old knowledge, not from introducing new proposals and checking for consistency with the old. A new proposal which is inconsistent with the old knowledge is not necessarily wrong, it could be that the old knowledge is wrong. — Metaphysician Undercover

I believe that there must be ingredients in the process of knowledge discovery that are fundamentally unknowable, because if we could know them, then we could even systematize the discovery of new knowledge, while this is fundamentally not possible. Therefore, every attempt at trying to harness the process is bound to fail. Hence, determining errors in the old knowledge cannot possibly be the main ingredient in the knowledge-discovery process either. For example, they did not start building the first computers because there were errors in the old mechanical calculators that preceded them.

I think it is possible for us to have a stimulating conversation about the outcomes of individual lives within our current economic and political system. — rlclauer

As an individual, the national system that you face, is a given. You cannot hope to change it. On the other hand, there are 200+ such national systems. In terms of what matters to me, at least 100+ of these national systems work absolutely fine.

Therefore, at a systemic level, it is much more important to me that one such national system cannot impose their possibly misguided views elsewhere.

In terms of geopolitics, it probably means asking powers like Russia and China to be much more confrontational with the USA, as to force them to stay more within their own national borders.

What we certainly do not want, ever again, is a repeat of removing arsehole Gaddafi in Libya in 2011, resulting in over two thousand militia still shooting at each other and vying for power, almost nine years later.

In Libya, the shelves are still empty in the supermarkets, there are shortages of everything, and nobody has a real job except for shooting other people. Can we, please, urgently dig up Gaddafi's dead body, resurrect him, and put him back on his throne?

When the two sides of a right angle are of equal length, the hypotenuse is irrational. Therefore the Pythagorean theorem as a first principle of geometry is deficient. Pythagoras himself grappled with this problem, and the fact that he could not resolve it bothered him. That the hypotenuse remains irrational indicates that the Pythagorean theorem remains unproven, just like the value of pi remains unproven.

If the rules are arbitrarily chosen then why choose a rule which results in the contradiction which is an irrational ratio? The fact is that the rules are not really chosen arbitrarily, they are chosen for purpose, pragmatics. The circle is useful, and pi is the result of the rule which creates the circle. The right angle is useful for making parallel lines, and the Pythagorean theorem is the result of the rule which creates the right angle. That each of these results in an irrational ratio indicates that they are lacking in truth and reality, despite the fact of being very useful. — Metaphysician Undercover

Irrational just means that a number cannot be reached by merely applying the standard arithmetic operators (+ - x /) to integers. So, if Z are integer numbers ...,-3,-2,-1,0,1,2,3,4,5 ... then you can see that this domain is nicely closed under addition, substraction and multiplication, because all results are again members of that domain. Example, 3+5*2 = 13. So (Z, {+,-,*)} is closed.

This algebraic structure (Z, {+,-,*)} is called a "ring".

This structure is not closed under division. For example, 2/5 or 1/3, are not members of the domain. If we adjoin the inverses of these integers, ... -1/3, -1/2,1/2,1/3 ..., then it closed. We call that resulting domain in which the inverses are adjoined, the rationals Q, and the closed algebraic structure (Q, {+,-,*,/}) where division stays within the closure, a "field".

For the calculation of the hypothenuse, you can see that mere field operations ("arithmetic") are insufficient. If a and b are the sides of the right angle, then the hypothenuse c = √(a²+b²) is not necessarily a rational, even if a and b are rationals. Only numbers produced by field operations on Q are guaranteed to be rationals. In other words, Q is closed under arithmetic but not under square root computation. So, c is not necessarily a member of Q, the rationals. In general, you will need to adjoin a radical field extension to the rationals Q in order to compute c.

So, the Pythagorean solution is "irrational" in a sense that it lies in a radical field extension of the rationals Q. Such radical field extension is then again closed under arithmetic.

Algebraic numbers are the domain that contains the rationals and all possible such radical field extensions, and is therefore closed under the n-th root operation. The algebraics are also a "field" that is irrational (meaning: completely contains the rationals Q).

Still, the algebraics are not enough when you look, for example, at the roots of polynomials with rational coefficients. You will need to keep adjoining additional field extensions if you want to close the splitting field. For example, you will at the very least need to add i=√-1. From fifth-degree polynomials on, you are not even guaranteed to stay within the algebraics. That is the gist of the Abel-Ruffini theorem, which is an important result in Galois theory. Polynomial splitting can then result in roots being non-algebraic real numbers (or complex numbers).

So, in this context, "irrational" just means that the problem cannot necessarily be solved by using basic arithmetic, but that it may requiring adjoining to the rationals Q, other numbers produced with more complicated operations.

It took until the end of the 19th century before the dust more or less started settling on these things. Before that, they did not understand these algebraic structures particularly well.

Actually, telescopes came after it was theorized that the earth revolved around the sun, and not vise versa, so understanding the heliocentric nature of the solar system was not the result of telescopes. The idea was floated around 2500 years ago, but the planets were given perfect circular orbits according to the principles of Aristotelian metaphysics. The assumption of perfect circles resulted in inconsistencies which could not be reconciled until Copernicus. The point though, is that metaphysical theory preceded the fine tuning observations which were required to adjust the theory. — Metaphysician Undercover

There is indeed a massive and intractable issue with the issue of discovery of new knowledge.

Existing knowledge cannot possibly be the main ingredient in the discovery of new knowledge, because in that case humanity would never have discovered any knowledge at all, or else, discovered all possible knowledge already.

Hence, the discovery process of new knowledge cannot possibly be justifiable as knowledge. We simply do not know how to discover new knowledge, and we can certainly not justify how we managed to do it anyway. It is to an important extent the result of non-knowledge mental faculties and possibly also fundamentally unknown environmental inputs.

Gödel's first incompleteness theorem also provably dismisses the idea of running through all possible well-formed formulas as to question a knowledge machine whether the formula is provable or not. For example, in the language required to axiomatize the existence of numbers, it is possible to produce formulas that are logically true but impossibly provable by the knowledge machine. So, if you enumerate the well-formed formulas in that language (which happens to be first-order logic), from first to last, the knowledge machine will run into examples of formulas of which the provability is simply undecidable.

So, it is just not possible to run new candidate knowledge claims through a knowledge machine filled with existing knowledge to check if these new claims happen to be justifiable. Gödel proved that this is not a legitimate knowledge discovery procedure. We will undoubtedly have to keep doing it with leaps and bounds, through serendipity, trial and error, and what have you, to slowly, gradually, and painstakingly, but surely, acquire new justifiable knowledge claims.

Of course, that's the nature of knowledge. Proceeding from the first principle has a similar problem,. There's no infinite regress, just some degree of uncertainty within knowledge, such that knowledge is forever evolving as we move forward. — Metaphysician Undercover

That is only the nature of falsificationist knowledge. That is absolutely not the nature of axiomatic knowledge. The Pythagorean theorem was provable 2500 years ago. It still is provable today. The same holds true for Thales' theorem. It is as provable today as 2500 years ago. Once provable, always provable. Hence, that particular view on the nature of knowledge is epistemically completely incorrect for axiomatic knowledge.

OK, now the point is that someone must determine the rules, the law. It makes no sense, to argue as you do, that all respectable knowledge proceeds from first principles in an axiomatic way, because this neglects the fact that someone must determine the principles, in the first place, from which the axiomatic knowledge will proceed. — Metaphysician Undercover

For mathematics, these rules are arbitrarily chosen. You can find a good explanation of how it works in the Wiki page on the Brouwer-Hilbert controversy:

Hilbert's axiomatic system – his formalism – is different. At the outset it declares its axioms. But he doesn't require the selection of these axioms to be based upon either "common sense", a priori knowledge (intuitively derived understanding or awareness, innate knowledge seen as "truth without requiring any proof from experience"), or observational experience (empirical data). Rather, the mathematician in the same manner as the theoretical physicist is free to adopt any (arbitrary, abstract) collection of axioms that they so choose. Indeed, Weyl asserts that Hilbert had "formaliz[ed] it [classical mathematics], thus transforming it in principle from a system of intuitive results into a game with formulas that proceeds according to fixed rules". So, Weyl asks, what might guide the choice of these rules? "What impels us to take as a basis precisely the particular axiom system developed by Hilbert?". Weyl offers up "consistency is indeed a necessary but not sufficient condition" but he cannot answer more completely except to note that Hilbert's "construction" is "arbitrary and bold". Finally he notes, in italics, that the philosophical result of Hilbert's "construction" will be the following: "If Hilbert's view prevails over intuitionism, as appears to be the case, then I see in this a decisive defeat of the philosophical attitude of pure phenomenology, which thus proves to be insufficient for the understanding of creative science even in the area of cognition that is most primal and most readily open to evidence – mathematics."

If you assume that all of the first principles for all divisions of knowledge have already been produced, this contradicts your original statement above, that we cannot know it's really "true", and therefore we must keep searching, in an endless way. You can't argue both sides of the contradiction. — Metaphysician Undercover

What has gradually emerged are epistemic knowledge-justification methods, whereunder axiomatic, scientific-falsificationist, and historical. Each of these epistemic methods generates an epistemic domain around it, i.e. a database of knowledge that can be justified with it. I cannot see what else you could be looking for, because there is nothing else, and there hasn't been for 2500 years.

What is bullshit is your claim that there has been no progress in metaphysics in 2500 years. — Metaphysician Undercover

So, then where is that elusive progress visible? Any link?

Do you think that human beings developed the current knowledge of the solar system, and the rest of the universe, by following the principles which were accepted 2500 years ago? — Metaphysician Undercover

Science is falsificationist. It is an epistemic domain of knowledge justified by experimental observation/testing. The initially hypothetical knowledge was very often stumbled upon, through serendipity, trial and error, and sheer luck. Systematic testing obviously always follows much later. So, yes, a better understanding of the solar system and other parts of the visible universe took a lot of observation. In fact, it first took quite a bit of haphazard progress in optics and construction of telescopes just to be able to observe these things in sufficient detail. So, yes, if they had had proper telescopes 2500 years ago, they would obviously have seen it too. It wasn't a problem of following the wrong principles at all.

You're smart in one area and a delusional bullshit artist when you talk to me. You got busted. Give it a rest. — fishfry

Is it about the uncanny similarity I mentioned between the size of finite calculation fields (prime powers only), yielding gaps in between, and the size of infinite calculation fields (also prime powers), and also with gaps in between (the continuum hypothesis)?

Why would that similarity be delusional bullshit? I have never said that I have proved anything about it.

You know, like so many people in the academia, whom Nassim Nicholas Taleb liberally calls IYI ("Intellectual Yet Idiot"), you seem to have an overly strong attachment to useless credentialism, that does not impress anybody who works in technology.

Being smarter than anybody else, means that you can do more. It does not mean: Being good at looking for things you disagree with in order to disparage other people. The problem is, of course, that there is absolutely no other benchmark for merit in the academia besides the ridiculous pieces of paper and citations that they distribute to each other.

Even if I think that you made a mistake somewhere -- probably debatable -- I will not easily call you "delusional" for it. It is rather your irresistible desire to put other people down, that is so stupid.

Is it maybe because elbowing your way through life is such an important requirement in your professional environment? Is putting down others more effective with a view on getting ahead, than doing something remarkable yourself? In that case, I do understand you.

Myself, I prefer working in an environment where people are free to develop ideas and to make mistakes without having to deal with that kind of obnoxious negativity.

LOL. One (you, me, anyone) would need a Ph.D. in set theory and several years of specialized postdoc work just to read what he's done so far. You keep making this laughable claim that you'll deign to read his work when he's done. You're embarrassing yourself. — fishfry

I just read Woodlin's conclusion: "unfinished work". The details of Woodlin's work are indeed "hard" and take a long time to understand. So, yes, I will skip the thing.

Still, Woodlin's work is certainly not the hardest stuff I have ever run into (or started reading). For example, I consider ZK-STARK theory ("Zero-Knowledge Succinct Non-Interactive Argument of Knowledge") to be much, much harder. The core ZK-STARK tutorial is subdivided in the following topics:

• Homomorphic Hiding
• Blind Evaluation of Polynomials
• The Knowledge of Coefficient Test and Assumption
• How to make Blind Evaluation of Polynomials Verifiable
• From Computations to Polynomials
• The Pinocchio Protocol
• Pairings of Elliptic Curves

Especially "Blind Evaluation of Polynomials" is pure genius but I find it really, really "hard".

Some of the guys who wrote this stuff have Ph.D's but almost none of the programmers does. For example, Vitalik Buterin successfullly reimplemented ZK-SNARK in the Ethereum code base, and he never even went to university. He is in his early twenties, and did not have the time for that, because Vitalik was too busy being some kind of cryptocurrency pop star.

Quite a few of the Ph.D crowd will say that they understand the stuff, but when push comes to shove, they will not be able to implement it, not even to save themselves from drowning. So, don't show me the Ph.D piece of paper. Show me your source code instead. If you really want to know about Ph.D's then read the article "Why can't programmers ... program?" Seriously, as Linus Torvalds said: "Talk is cheap, my friend."

Another reason why Vitalik does not have a Ph.D in set theory is because he really does not need one to wipe the floor with you in set theory, if he so desires. Furthermore, nowadays, you are much better off doing some advanced crypto, if you want to be some kind of math super star.

So, yes, if you want to discuss "hard" stuff, then let's at least pick something I can make money with.

The market capitalization of the ZCash cryptocurrency is around 400 million dollars now. So, at least we would know what we are doing it for. There is just one catch. The ZCash people are working on the upgrade to ZK-STARK now. That is what has really discouraged me from "completely" reading up on their theory, and figuring out the related function libraries. They are going to throw it away!

So, you may think that advanced set theory is too "hard" to read, but sorry, it is a walk in the park compared to what we do, when we do what it takes to get the software to run. Furthermore, the real question is: Is it really worth it?

By the way, the only thing that postdocs (as well as associate lecturers) have in common is that they can't pay their bills (from their food stamps).

Again the reality of those concepts are not the job of mathematics, it is the consequential load in those rule following games that is mathematics. — Zuhair

Totally agreed.

There is even an interesting paragraph on this principle in Wikipedia:

A philosophical defeat in the quest for "truth" in the choice of axioms

(Note: I personally consider it to be a victory.)

For good measure, and in order to stamp out the constructivist heresies that keep flying around, I propose to dig up and resurrect Luitzen Brouwer's dead body and to ritually kill him again. The problem is that Satan never dies completely; and unfortunately, neither does his notorious accomplice, Luitzen Brouwer, the worst constructivist heresiarch in the history of mankind.

I corrected some of your errors and you started making wild extrapolations of things you didn't understand. — fishfry

Well, I have said that Woodlin's work is surely interesting, but that Woodlin himself admits that it is not finished, and that I will read up on the details when he does finally finish his work. In the meanwhile, I agree that I refuse to read up on the difference between cardinals that are "extremely large", "super huge", or otherwise "incredibly out-sized". These things are obviously not the same! Ok. happy now?

What in God's name difference does it make if JTB or TB is true? Who has ever cared about that other than a few people with too much time on their hands? Who cares how we define knowledge? This is probably what the professor in the OP was talking about - this is the kind of crap philosophers are forced to waste their time on. — T Clark

Not sure.

I keep getting irate remarks from @fishfry in another thread because I refuse to read up on the nitty-gritty details of the cutting-edge research on the Continuum Hypothesis (in math). He seems to insinuate that my point of view -- I have to draw the line somwhere, don't I? -- is pure evil.

So, let's say that there is some kind of (relatively small) fan club for bleeding edge research on JTB, while everybody else clearly does not give a flying fart. Isn't that the case for almost *everything* ?

By the way, who the hell would ever have thought that there is complete fan club with greater-than-life pop stars to be found in the "epistemology of randomness"?

Taleb and Nobel laureate Myron Scholes have traded personal attacks, particularly after Taleb's paper with Espen Haug on why nobody used the Black–Scholes–Merton formula.

People are even still siding in the notorious insult fest between David Hilbert and Luitzen Brouwer, even though the antagonists are now dead already (for decades). I surely side with David Hilbert for 100%, and I keep throwing vitriol at Luitzen Brouwer, who in my opinion, is the accomplice of Satan.

That's contradiction, "first" means first, the possibility of infinite regress is therefore excluded. — Metaphysician Undercover

But how do you know it is truly "first"? You do not. So, you will keep trying to find the really "true" first that comes before the current first. It just keeps going on. Ad nauseam. That is why it does not work.

Unless you can justify this claim, it's nothing more than an opinion of an uneducated person. — Metaphysician Undercover

Knowledge is a gigantic database with lots and lots of theories and theorems.

I do not make any attempt whatsoever at memorizing that database.

So, if you mean that I have not read 99% of the database of existing knowledge, I totally agree. I even pride myself on not having done that. I only loosely remember, if even, the few things I accidentally have run into, usually for very arbitrary reasons.

I am more than happy with that, because my ambition is not to become some kind of redundant database of knowledge who in vain tries to be a very imperfect replacement for tools like Google Search or Wikipedia.

People really need to develop a purpose in life that is different from that, because their plan is otherwise bound to fail. So, are there a lot of things that I do not know? Yes, of course, and I am very proud of that.

I gave you the example, moral ethics ... I see, morality is nonsense to you. — Metaphysician Undercover

The most widespread and successful approach to morality is what the three offshoots of second-temple judaism propose, i.e. religious law.

As far as I am concerned, the epistemically soundest version of the religious-law morality method can be found in usul al fiqh, "Principles of Islamic jurisprudence". It is a gigantic library of innumerable publications.

Read up on it, and then you will understand that what you are doing in the realm of morality, i.e. "metaphysics", is just un-methodical bullshit. Seriously, that is why there has been no progress whatsoever in metaphysics for over 2500 years. That was to be expected, because there is simply no logic in that madness.

I'm always up for a chat about set theory but this ain't it. — fishfry

I am not a researcher in set theory. It is obvious that there are numerous people trying to advance the field, but I am not one of them.

If Woodin said that he's "not finished," he means he's not finished with his decades-long research project. — fishfry

The Continuum Hypothesis (CH) has been an outstanding issue for over a century now. Woodin is trying out an interesting direction to prove it. That's about it, I guess.

He's got plenty of published proofs, he's one of the top set theorists in the world. — fishfry

Of course, Woodin has published commendable work and numerous proofs, but not one for CH. Well, not yet ...

our comment that there's nothing on Wikipedia because he doesn't have a finished proof is, I apologize for my directness, laughable. — fishfry

Well, that's the way it works ...

That's your strategy. Avoiding knowledge. — fishfry

There is no point in me trying to learn the nitty-gritty details in cutting-edge set-theoretical research, because it is not my job to try to advance the field. ZFC is endless. There is obviously no limit to the number of theorems that can be developed in it.

But then again, the article you mentioned yourself, argues that cutting-edge ZFC research is not going to help with solving CH:

It is the same with the continuum hypothesis: we know that it is impossible to solve using the tools we have in set theory at the moment. And up until recently nobody knew what the analogue of a ruler with two marks on it would be in this case. Since the current tools of set theory are so incredibly powerful that they cover all of existing mathematics, it is almost a philosophical question: what would it be like to go beyond set-theoretic methods and suggest something new? Still, this is exactly what is needed to solve the continuum hypothesis.

I find CH to be interesting, but it is certainly not something I encounter in a professional context. So, why spend time on that and not on something else?

all the foundational level work has been done — RogueAI

That is not true!

For example, defining knowledge as a justified true belief is clearly unsustainable.

Edmund Gettier famously breached the stalemate in 1963 with his counterexample cases. The entanglement phenomenon also decisively breaches the classical JTB definition. The problem is now completely up in the air, even on the empirical side of things.

Furthermore, only empirical knowledge could possibly ever be correspondence-theory "true" and therefore JTB knowledge. Axiomatic fields such as mathematics, which are never correspondence-theory "true", are not knowledge in that approach. So, what are they then?

I think he was right. The original stuff has already been thought of. There's been too many smart people for anyone to have missed anything fundamental by now. We need new perspectives. — RogueAI

Well, you cannot produce original insights by merely reading and rehashing the classics!

That is trivially obvious.

You will, instead, need to bring some interesting experience from looking at what people do in a particular practical field around you. The collection of philosophyOf(X) fields is much larger and ultimately also much more interesting than just X=philosophy itself.

A good example of an excellent epistemologist of X=randomness, is Nassim Nicholas Taleb.

I have read every single one of his Incerto books:

• Fooled by Randomness (2001)
• The Black Swan (2007)
• Antifragile: Things That Gain from Disorder (2012)
• Skin in the Game: Hidden Asymmetries in Daily Life (2018)

In my opinion, Taleb's books are pure genius.

You see, Taleb's experience is in finance. The finance industry uses and abuses ceremonial rituals in mathematics and science as smoke and mirrors. In reality, they just sell snake oil. Taleb also points out that the finance industry is not the only industry doing that, i.e. repackaging the superficial appearance and rituals of solid mathematics and serious science into costly snake oil. His Incerto books are fantastic, if only, because they remind us of the fact that we are entirely surrounded and outnumbered by dangerous gangs of deceptive liars.

NNT has his own subreddit of "groupies" discussing his every tweet or other public appearance. He has an impressive fan club ...

But metaphysics is reasoning toward first principles, not reasoning from first principles. — Metaphysician Undercover

That is even worse.

If you reason toward first principles, you will look for the principles underlying these first principles, and again, ad nauseam. It obviously leads to infinite regress. That is why this particular direction is forbidden in axiomatic systems. It only works by picking an unjustified starting point and reason away from them, i.e. exactly in the opposite direction. The goal is then to justify conclusions from that starting point, without falling into the trap of trying to justify the starting point itself.

That is why the axiomatic starting point in mathematics, the axioms, are fundamentally arbitrary, while in morality the starting point, the categorical imperatives, are necessarily "revealed".

But to study morality, as a field of study within philosophy, is a process by which we seek to determine those rules. — Metaphysician Undercover

Justifying the starting-point rules is an exercise in infinite regress and futility. Can you give even one example of where an approach like that has worked?

but the metaphysician reasons toward determining first principles. — Metaphysician Undercover

The metaphysicist is wasting his time, simply because the direction of reasoning is necessarily incorrect. What the epistemologist does, however, makes much more sense.

Instead of looking at the real, physical world, he looks at the abstract, Platonic world of knowledge and tries to discern if particular patterns emerge. The scientist does that with the real, physical world, while the epistemologist does that with the abstract world of knowledge. An epistemic pattern then demarcates an epistemic domain. A good example is how Karl Popper successfully managed to demarcate the epistemic domain of scientific knowledge by requiring science to be generated by falsificationist activity.

Epistemology really works, while metaphysics is nonsense. We know that for a fact, because after 2500 years of metaphysics, it has never produced anything else but nonsense.

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Empiricism is about correspondence and Mathematics is about consequence. — Zuhair

Agreed. It establishes science and mathematics as epistemic domains instead of subject matters. They are indeed not about anything. They are knowledge-justification methods.

Logical positivist used the verificationism principle to regard metaphysical statements as meaningless, would you go along that belief ? — Wittgenstein

No, I don't. I just share the same conclusion, but for different reasons.

I think that presuppositionism about the real, physical world is ineffective, because the real-world principles are often testable, and should actually be testable. So, if something is testable, why don't they just test it? In that sense, I believe that real-world knowledge should rather be falsificationist. Karl Popper did a great job in pointing that out in "Science as falsification". Permitting real-world presuppositionism will invariably lead to non-knowledge, snake-oil scams. So, that is a big no, no.

Reasoning from first principles only makes sense in the context of abstract, Platonic worlds, simply, because we can actually know their first principles, i.e. their construction logic. For example, the axiomatic method certainly does an excellent job in mathematics; but it also does an excellent job in morality, where axiomatic derivation from basic rules is also the method of choice.

Reasoning from first principles in the context of the real, physical world looks like a serious epistemic mismatch to me. That is why I reject the practice of metaphysics.

We find out what axiomatic systems give what conclusions, but notice that the conclusions that we desire are the motivating feature in this diagram. — fdrake

There seems to be an entire area of research on axiomatic subsystems of second-order arithmetic (Z2). It is apparently about crippling Peano arithmetic and check what's left over. Presburger and Robinson are examples of this, but there are also other variations.

Pure formalism just gives you the black arrows, it does not give the sense of mathematical progress through the articulation and codification of ideas, just the dynamics of symbols, as if the ideas motivating them were completely irrelevant. Another way of putting it: formalism is just what we invent to get to where we need to go. — fdrake

I think that they do not care, for example, in Z2 research. They're just interested in what the effect is of the crippling on the resulting arithmetic. For example:

Skolem arithmetic is the first-order theory of the natural numbers with multiplication, named in honor of Thoralf Skolem. The signature of Skolem arithmetic contains only the multiplication operation and equality, omitting the addition operation entirely.

It is actually interesting stuff. I am surprised that this botched version of arithmetic still seems to work.

I in opposition to your terminology prefer to use the term truth to denote another context which is quite different from the "correspondence with reality" context, and that context is what I've labeled as "consequential truth", you are free not to call it truth, you may term it as "consequentiality", or "consequential processing" which are fair enough. — Zuhair

"Consequential truth" sounds very much like Coherence theory of truth.

In epistemology, the coherence theory of truth regards truth as coherence within some specified set of sentences, propositions or beliefs. The model is contrasted with the correspondence theory of truth. A positive tenet is the idea that truth is a property of whole systems of propositions and can be ascribed to individual propositions only derivatively according to their coherence with the whole.

"Coherence" in mathematics is obviously axiomatic (=reductionist). Fundamentally, I object to this view because it turns "truth" into some kind of calculation. Any such calculation can be not just arbitrarily faulty, but also fundamentally misguided.

Perhaps the best-known objection to a coherence theory of truth is Bertrand Russell's. He maintained that since both a belief and its negation will, individually, cohere with at least one set of beliefs, this means that contradictory beliefs can be shown to be true according to coherence theory, and therefore that the theory cannot work.

As Bertrand Russell argued, naive approach to coherence would clearly lead to contradictions.

However, what most coherence theorists are concerned with is not all possible beliefs, but the set of beliefs that people actually hold. The main problem for a coherence theory of truth, then, is how to specify just this particular set, given that the truth of which beliefs are actually held can only be determined by means of coherence.

At the same time, a selective approach is not really possible, because it is automatically circular.

I think that the better term for "consequential truth" in mathematics is "provability". It captures much better the status of mathematical theorems. If a theorem necessarily follows from a set of rules, then the theorem is "provable from" these rules. Why use the controversial term "truth" instead of "provability"? Especially given the fact that the rules from which a theorem is proven, do not need to be "true" in any sense ...

If by polymorphism you mean that truth can take different forms, I disagree, because on my definition, "truth" is defined to accommodate all forms, as qualified, because that is the best it can do. — tim wood

Polymorphism is first and foremost a language thing. I am not sure if it exists in the real, physical world. Maybe it does somewhere that I am not aware of. Polymorphism is obviously all over the place in software, like for example in duck typing:

Duck typing in computer programming is an application of the duck test—"If it walks like a duck and it quacks like a duck, then it must be a duck"—to determine if an object can be used for a particular purpose. With normal typing, suitability is determined by an object's type. In duck typing, an object's suitability is determined by the presence of certain methods and properties, rather than the type of the object itself.

So, yes, the practice is allowed, also in language in general (such as natural language), but the ever-present danger is: ambiguity.

And I cannot think of a fundamentally ambiguous situation. — tim wood

It is not necessarily the real world, or its situations, that are ambiguous. However, language definitely is: Every bug is an ambiguity and every ambiguity will sooner or later lead to a bug. The losing war against bugs is in reality one against ambiguity.

Any form of reflection on the "world" or "states of affairs" concerns facts, and facts are always historical. Are you prepared to assert that all facts are true - without some weird question-begging qualification? — tim wood

Yes, but subject to the scrutiny of the historical method.

My own definition of truth, fwiw, is that "truth" is an abstract term that simply means that the proposition in question complies with an appropriate standard in being true, while being entirely agnostic as to what that standard is. — tim wood

Well, polymorphism is obviously permissible as long as it does not lead to fundamentally ambiguous situations ...

imagine that you have an idea of a structure you want to capture the behaviour of, and you have the behaviour — fdrake

A weird aspect of math is that "behaviour" means, "respectful of particular invariants", such as e.g. :

a ⊕ b = b ⊕ a

commutativity or so. In (software) programming, it means being able to execute a particular function, e.g.

a->f(b)

where "a" is capable of looking up a function "f" and apply it to a given argument. I personally find programming with invariants to be much harder than with functions ... but ultimately probably also more powerful.

If ever you've tried to axiomatise a structure you'd see that there's a reciprocality between the structure's concept and its mathematical definition. — fdrake

With the term "structure", do you mean structure as in "algebraic structure", i.e. a set with a collection of operations? Such as a group (K,+) of set K and the addition operation?

Saying that it is a "group" automatically attaches a set of invariants. If you add enough invariants to the structure, i.e. you may use up all your degrees of freedom, then indeed, at some point there will only be one candidate definition that fits the bill. It could, for example, leave only one K possible. You could obviously also over-specify and propose the structure of something that cannot possibly exist.

I think that you can treat K as an unknown, or even one of the operations (why not?), but that will probably lead to specifying a non-trivial higher-order logic problem.

Are you trying to program a structure by attaching invariants?