For some reason revealing of my own unacceptable prejudice I'm slightly offended that you think I'm American. Anyway, I'm English, Our biggest problem is the BNP, UKIP etc., but the problem of Trump I see as an example, not an exception, and the British response has been instrumental. We've basically said that we don't want him over here to speak, that nothing he's got to say is of any interest to us. I think that's a very powerful expression of the contempt in which we hold his views, much more powerful than letting him over here and debating them, as if they had any kind of legitimate reasons that might require some thought. — Pseudonym

I think that's exactly the correct response in that case. I hope it lasts. Do you think May will give in and invite him over at some stage, despite the unpopularity of such a move with the British people?

For the first part of your post - I think your assessment of our respective positions is correct. We seem to agree on aims, but disagree on methods - at least as far as local talks and demonstrations go. And we seem unlikely to persuade one another. I hope you turn out to be right and I turn out to be wrong, because that will mean that your 'de-platforming' efforts have been successful in diminishing the influence of the UKIPs, BNPs and Milo Yiannopoulos's of this world..

I'm afraid I don't know. I'm hoping somebody that knows about international law will chime in here. I'd like to know the answers too.

hence invalidating Kant. — Agustino

I don't agree that it does, but I was wondering who might be an example that strong view that Wayfarer mentioned some people holding. So now I know.

You keep repeating that Euclid's parallel postulate is not intuitive, but you don't explain why. — Agustino

If I could explain it, it wouldn't be an intuition.

The other postulates seem obvious and undeniable to me. That one doesn't. I suppose it must be just the way my brain's wired.

At least I can tell you why it took so long to discover the other geometries though. It's because it wasn't just a question of removing the parallel postulate. It needed to be replaced by something, otherwise we're taking away too much. In fact, what was needed was a complete re-axiomatisation, starting with a completely new set of axioms that does not resemble the existing ones at all. In fact a completely new language was needed, involving things called manifolds, vector spaces, tensors and metrics.

That was a very difficult task, and needed to wait for some extremely clever people to first realise that's what was needed, then secondly work out how to do it.

I'm struggling to find an example from history which demonstrates the effect you're claiming, perhaps you could provide the examples you're working from? — Pseudonym

For a start, there are plenty of people saying that Trump got elected from people reacting against what they saw as an excess of authoritarian political correctness, incorporating such things as harrassing people that express unpopular views. Even if that's only a tiny bit true, the effects are enormous - apparently it could even end up in a nuclear holocaust.

I personally seem to come across plenty of people who are not particularly committed either way, but speak very disparagingly when they see aggressive demonstrators on the TV news, then keep referring back to it at regular intervals from then on.

You say the above is irrational. Well, yes, most people do not vote rationally - as I said, not like Chidi Anagonye. Ask any spin doctor.

Lets say the students behave and let the person speak, some academic responds in the media rebutting his racist claims (though what would have prevented him from doing so anyway I don't know but we'll skip over that for now). What difference would that have made to your voter? — Pseudonym

Then the issue would not have moved them to vote against whatever cause the students support. So they will vote based on some other (quite possibly irrational) consideration. But the bias against the progressive cause has been removed, so the expected number of votes against progressivity has reduced. That's a win for the progressive camp.

what's going to happen when the racists speaks, tells everyone how badly treated white minorities are in some ghettos, how positive discrimination is robbing white people of jobs, how white girls can't even walk the street in areas dominated by immigrants? — Pseudonym

The voter will probably never hear what the racist says, because they didn't go to the rally, and the rally won't make the TV news, because it was only the violent demonstrations that made it newsworthy. The demonstrators were essentially providing free publicity for the racist's cause.

People are not so impressed by dignified protest that they're going to turn away from the persuasive and powerful rhetoric that's saying exactly what they want to hear just because the opposition to it are well-behaved. — Pseudonym

We differ there. I think people are impressed by dignified protest. I think of Gandhi and Martin Luther King. And Nelson Mandela only became an international hero after he had been in jail long enough, and conducted himself in such a dignified manner, that people had forgotten he was arrested for arms offences. In Northern Ireland the most notable phenomenon leading up to the Good Friday agreement was not the violence of the IRA and UDA, but the increasing size and prevalence of peace marches.

I don't think Rosa Parks would have had nearly as much of an impact if, when arrested, she had started screaming, striking out at and spitting on the police officers that led her off the bus.

But it just occurred to me that maybe you're American (apparently many people on here are). If so then the biggest platform problem you've got is that your head of state is a fascist. So he can get horrifically mean and discriminatory views on the national news simply via twitter. I think that's a much bigger problem than a few white supremacist rabble turning up to a rambling diatribe at a lonely lecture hall of some university. But I'm afraid I have no suggestions for 'de-platforming' your leader other than to work towards (1) reducing his power via a loss of his party's majorities in Congress at the mid-term elections, and (2) removing him completely in 2018.

The "criticism" was simply that if indeed he believed the axioms of Euclidean Geometry were metaphysically necessary, then Non-Euclidean geometries seem to falsify this notion. — MindForged

Fair enough. I had assumed - wrongly, it now seems - that you were aligning with the group that @Wayfarer identified in this post ( ) that assume the discovery of non-Euclidean geometry undermines Kant’s understanding of a priori truth. If all you are suggesting is that Kant may have had a wrong idea about the necessity of the parallel postulate, then you are not adopting the assumptions of that group. The suggestion seems not to damage Kant's thesis at all, and I do not argue against it.

Since most people in Kant's time believed the parallel postulate was necessary in order to be able to do geometry at all, it is no adverse reflection on Kant, or on the Critique of Pure Reason or the usefulness of the Transcendental Aesthetic, if he believed that as well.

Unlike spacetime curvature, which is coordinate-independent, the curvature of space (strictly, of constant-time hypersurfaces) varies according to the coordinate system being used to make measurements.

I have just been informed by an impeccable source that there are coordinate systems in which space near a large mass like the Earth is locally perfectly flat. Here's the wiki page that describes those systems.

Problem solved! Immanuel Kant has been vindicated. X-)

As an aside, my source pointed out that, even in the Swarzschild coordinate system that is more typically used near a planet, the spatial curvature near Earth would be about one part in a billion, and probably not possible to detect with current equipment.

Another way of arguing: you could say that our intuition of space is actually non-Euclidean (or whatever happens to be the correct geometry of space, supposing non-Euclidean geometry is superseded), and Euclidean geometry was merely an empirical concept of that form. — Moliere

Nice!

Looking up Euclid's axioms on wiki, I noticed something interesting. Here's wiki's translation of them:

Let the following be postulated":

1. "To draw a straight line from any point to any point."
2. "To produce [extend] a finite straight line continuously in a straight line."
3. "To describe a circle with any centre and distance [radius]."
4. "That all right angles are equal to one another."
5. The parallel postulate: "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles." — wiki version of Thomas Heath's translation of Euclid

Unless I'm missing something, the 5th postulate would also be true for an elliptic surface, such as the surface of a sphere. In order to exclude elliptic geometries, the words ',and not on the other side' would have to be added at the end of the sentence.

The same applies to the alternative version of the postulate given in the following section of the wiki article:

In a plane, through a point not on a given straight line, at most one line can be drawn that never meets the given line.

It seems to me that 'at most' needs to be changed to 'exactly' in order to exclude elliptic surfaces.

I feel I must be missing something. Can anybody help me find what it is?

What exactly is the criticism? That he didn't discover the existence of elliptic and hyperbolic geometries in 1781? That's a pretty harsh standard, since nobody else came up with them until at least 1813.

Or that he didn't at least understand that the parallel postulate was not necessary in order to have all the usual concepts of continuity, connectedness, insides, outsides, points, lines, angles, volumes, shapes? Again, nobody else knew that until 1813, so why should we have expected Kant to realise it?

The following is counterfactual, and hence unfalsifiable and otherwise empty, but no more so than the rest of the discussion:

I suggest that if a mathematician that Kant respected had discovered these things and had explained to Kant in 1781 that you can get all those things without the parallel postulate, Kant would have taken that on board and related his Transcendental Aesthetic to Riemann and his manifolds, rather than to Euclid.

I don't think one can say he was still correct in insisting EG as metaphysically certain — MindForged

What I'm pointing out is that the words 'Euclidean Geometry' have a different meaning now from what they had in the 18th century. In the 18th century they just meant Geometry simpliciter, because Euclid was seen as the father of geometry and was considered synonymous with it, and because no other sort of Geometry was known and people imagined no other sort was possible.

But now the term 'Euclidean Geometry' is used to refer to a subset of Geometry that excludes manifolds with curvature. To argue that Kant intended that meaning, without additional evidence, is to participate in an anachronism, using a meaning of the term that was not the meaning at the time it was used.

The only way to substantiate a claim that Kant was not just referring to Geometry generally (Riemannian Manifolds) is to find a quote where he specifically insists on the importance of the parallel postulate.

I see that @Agustino has just posted a new quote from Kant involving the sum of angles in a triangle, so I'll read that and see where it leads me. Since I find reading Kant really hard work, it'll probably be quite a while before I have anything coherent to say about it.

I'd love to put my idea to a test so I'd value hugely any examples of any situations where the model could or could not be applied as well as any questions. — BlueBanana

Two examples you might find interesting to contemplate:

1. I believe Antarctica is not 'owned' by any country. I think there's an international treaty under which a small number of nations have restricted control of certain portions of it, but that only extends to making certain scientific uses of it - nothing like the total control and permission to exploit that ownership is usually understood to imply. I don't know much about it, but it may be relevant to what you're asking.

2. I think there's some sort of international agreement that nobody can own the Moon, or any part of it. That may repay looking into.

Thank you for contributing an actual quote from Kant to the discussion. As is usual with me and Kant, I'm not very sure about what he means by it, but I do notice one thing, which is that although he sounds very definite - almost dogmatic - about the primacy of the notion of space, and does connect it to axioms - presumably axioms of geometry - he does not mention Euclid's parallel postulate, which is what distinguishes Euclidean from non-Euclidean geometries.

Before Gauss, Lobachewski, Riemann and others developed the notion of non-Euclidean geometry in the 19th century, people tended to regard Euclid's axioms as a job lot, which you accepted or rejected all together, not picking and choosing. Nobody could see a way of accepting some but not others and still coming up with workable, useful notions.

But now, thanks to those 19th and then 20th century mathematicians we have a good understanding of many different possible axiomatisations, all producing workable, useful spaces. The mathematics is called Differential Geometry and the spaces are called Riemannian Manifolds.

What is common between the different spaces/manifolds - Euclidean and Non-Euclidean - is that they all satisfy the axioms of a Riemannian Manifold, the key ones of which are:

- the space is connected, so that from any starting point you can get to any other part of the space via a continuous path
- the space is continuous, so going along a path you won't suddenly find yourself in a completely different, faraway part of space
- the space has a constant dimensionality that is a positive integer n. For our space n=3.
- between any two points there is a measurable 'distance', which is the length of the shortest path from one to the other, and these distances must:
* be non-negative
* are zero iff the two points are the same
* are symmetric, so that the distance from A to B is the same as the distance from B to A
* obey the triangle inequality, so that the distance from A to C does not exceed the distance from A to B plus the distance from B to C

The Riemannian Manifold axioms are enough to give a very strong notion of space as a three-dimensional area in which things are located and can move about. Further, it accords well with our intuitions of space - well, with mine at least!

My hypothesis, which may be dispelled by further direct Kant quotes, is that perhaps it's this more general notion of space notion that Kant was insistent on, not on a notion that added additional axioms to make the space Euclidean. The test would be whether Kant actually directly mentions Euclid's parallel postulate, or something equivalent to it.

Note that mentions of Euclid or Euclidean do not count, because in Kant's time those terms only indicated a reference to geometry generally, not to something that is distinct from Non-Euclidean space - a meaning that only arose in the 19th century.

In the Kant quote that Agustino kindly provided for us (link above), Kant only mentions two specific aspects of space, which are:

(1) the triangle inequality for distances; and
(2) that things can be 'inside, outside or alongside one another'

These are properties that are satisfied by any Riemannian Manifold, not just Euclidean ones. Perhaps he mentions the parallel postulate somewhere else, but he certainly does not do so in the above quote.

One last thing. The parallel postulate says that there exist pairs of straight lines that never meet, and that pairs that do meet only do so at one place. I, and generations of mathematicians before me, do not find that particularly intuitive, whereas Euclid's other axioms do seem intuitive. That's why people wondered for centuries whether that aximo was necessary in order to do geometry at all. Gauss's brilliance was to show that it wasn't.

Another aspect of the parallel postulate is that the three internal angles of a triangle must add to 180 degrees (or to 'two right angles' as Euclid put it). Again, I do not find this at all intuitive. In non-Euclidean geometry the sum of angles can differ - it is more than 180 for elliptical manifolds and less than 180 for hyperbolic ones.

Anyway, TLDR, sorry about that, My question is: did Kant ever specifically insist that Euclid's parallel postulate was part of our a priori processing of intuitions?

These are excellent admissions. — Thorongil

There were no admissions. You're making stuff up again.

My position has not changed one iota. The only thing that has changed is that maybe you are finally starting to realise that all the conclusions that you leaped to about what my position was were ridiculous and unfounded.

Why do you find it so hard to just say 'Sorry, I got it wrong.'

I'm not sure we can rely on geometry remaining Euclidean at human scales because if one takes general relativity into account then masses are warping spaces and causing gravitational geodesics that support our entire existence. — Perplexed

The curvature that gives us our fairly modest Earth gravity is a curvature of spacetime, not a curvature of space. The difference between the two concepts is crucial. IIRC, it is possible to have a spacetime that is curved but for which all spatial slices are flat. The curvature is only in the relation between space and time, not in the space itself.

Kant saw space and time as separate, so curvature in the relations between them, would have meant nothing to him, and could not contradict his 3D + 1D model* that is the Transcendental Aesthetic . Indeed, the maths needed to express that curvature had not even been developed when Kant was around. All that mattered was that spatial slices ('constant-time hypersurfaces') were flat, and that is extremely close to being the case in a site of very small gravity like the surface of the Earth.

* As distinct from Einstein's 4D model.

The implication cannot be seen because it is not there. You are imagining it.

You are never going to be able to conduct a sensible discussion if you keep on telling people that they said or meant something that they didn't. You are not a mind reader.

You can bold my words as much as you like. That won't make an accusation of 'racism' appear amongst them. 'Racism' is a concept that I find deeply unhelpful and avoid using wherever possible.

Identifying a harm that a person suffers is not the same as blaming somebody for that harm. If that's the way you look at the world then that's your misfortune, but it's just silly to assume that everybody else looks at the world the same way.

Good, progress. Notice these things have nothing to do with being white. — Thorongil

More made-up irrelevant nonsense. Nowhere did I say that police who shoot blacks are all white, or even predominantly white.

I suppose it's futile to hope that at some stage you'll actually start to engage with what people have written, rather than what your feverish imagination tells you they might have written.

Now you're just making things up:

Asians are being racially abused on buses at epidemic levels?

Nobody but you said anything about epidemic levels.

Sorry, but racism isn't anywhere near the most significant factor in why police shoot some black people

Nobody but you said that racism was a factor, let alone the most significant one. FWIW I think the major factors are fear, poor training and lack of psychological screening.

you attribute the mere fact of a black man being shot by a police officer to racism.

Find where I said that and quote it back to me, with link. You won't be able to find it, because I didn't say it. In fact I don't think I have used the word 'racism' at all in this thread, prior to this post (where I mention, but do not use it).

You are not listening to anything anybody says. You project onto their posts what you think somebody that disagrees with you might have said, and attack that instead.

Really? You think we should be glad that racial abuse of Asians, and police harassment and shooting of blacks occurs regularly? OK.

To be clear, do you really want us to understand that you believe that wishing something was different is 'prescriptive'? Does that also apply to when you wish recovery of a friend or family member from a horrible illness?

One counter-example is me. I used the phrase a few posts ago. I know that white privilege exists but, sadly, I have no prescriptions to offer to eradicate it, so I'm not advancing any ( not even 'implicitly'!).

Also, noted that when caught out making a ridiculous, insulting assertion, your response was not to apologise and correct yourself but to instead claim pedantry/

I wonder if those concepts could fit into the transcendental aesthetic since they're so far beyond the normal framework of space/time.

For me they seem to. When I visualise many-dimensional branes colliding in even higher dimensional space, I visualise hanging, wobbly two-dimensional sheets banging into one another in 3D space. The calculations will be different from the 3D case but for me the visualisation has to remain 3D (or at most 4D - I sometimes use time as proxy for a fourth spatial dimension) as I am not capable of visualising anything higher.

Isn't the issue that Kant elevated the postulates of Euclidean geometry to the level of a metaphysical certitude

Did he do that? I don't know. I never read the original, being a secondary-source kind of chap. I think we'd need to dig out a quote, both in the original German and a diversity of English translations, and analyse it to see whether we can reach that conclusion from it. My loose observation about Kant scholarship is similar to that commonly made about economics: If we put n Kant scholars in a room there will be n+1 opinions about what Kant meant by any particular passage he wrote.

not a single person who uses it fails to either implicitly or explicitly advance various prescriptions.

really? You were witness to every single time it was used, and knew exactly what each person that used it was thinking were you?

Bullshit on stilts. I can't believe people write this stuff with an apparent straight face. — Thorongil

Your cool eloquence, the dazzling logic of your shining prose, has persuaded me. What can I have been thinking to say what I did?

No I didn't (Well spotted!). I was going to heap special praise on him for that valiant attempt, but decided not to because I didn't want to embarrass him.

A quote from the essay that summarises the problem:

"....Geometry that fails to follow Euclid's assumptions is, according to Kant, literally inconceivable."

Frank Wilczek — Perplexed

Non-Euclidean geometry does follow Euclid's assumptions at the scales that are meaningful to humans. So there is no conflict.

A geometry that is not locally Euclidean would have to be fractal - infinitely wiggly - and most of us (certainly me!) find that inconceivable. I can do formal calculations about infinitely wiggly things (the Ito Process that is used as the basis of pricing most financial instruments is infinitely wiggly), but I cannot visualise them, and I have no intuitions about them.

Since Kant was tremendously intelligent, and not only understood physics but even published important papers in it, I have no doubt he would have very rapidly understood non-Euclidean Geometry and its usefulness if it had been around when he was alive.

I see no reason to suppose that it would undermine Kant's notion of the Transcendental Aesthetic, which is that we process raw inputs within a framework of three space dimensions and one time dimension. Kant would not have said that the framework could not be non-Euclidean, since the concept was not known in his day. He may well have mentioned Euclid, but that would have been because at the time Kant was writing, the name of Euclid represented all known geometry. Non-Euclidean geometry only became well-known after Gauss wrote about it in the 1813. So Kantian references to Euclid can be interpreted simply as unqualified references to geometry.

Being a great scholar and physicist, Kant, upon being introduced to non-Euclidean geometry, would have rapidly noticed that an essential feature of the non-Euclidean geometry that is used in physics is that Euclidean geometry is an extremely accurate approximation to the non-Euclidean geometry at non-cosmic scales. Since humans live, work and think in non-cosmic scales, I expect Kant would have been entirely comfortable with the notion that our in-built mechanism for arranging information is an approximation to a paradigm whose differences are only visible at scales that are beyond ordinary human experience.

Consider firstly the effect such images would have on a theoretically equivocal voter. What views would they already have to hold in order that such images would actually persuade them one way or the other? — Pseudonym

All they need have is a dislike of bullying, which is a very common dislike. Regardless of whether such protests actually are bullying, they look like bullying when shown on the TV news, and that's enough to turn many uncommitted voters against whatever it is that the protesters represent.

Most people do not make decisions in the way that Chidi Anagonye does.

What I don't get then, is what kind of weird reverse psychology do we imagine would cause them, on seeing how violently a group of students do not want a racist to speak, to think "well I wasn't so sure about racism before, but I am now". — Pseudonym

That is only relevant if there is no debate about whether the person giving the lecture is a racist, and the person accepts the label themselves. In the real world, that is almost never the case. A more likely interpretation would be

'Those students are screaming that XYZ is a racist and jostling people that are trying to attend the lecture. Those students look like horrible, aggressive bulllies, so I doubt they are believable. So it seems likely that their accusation that XYZ is a racist is false. So not only are they bullying him, but they are falsely accusing him of a horrible thing, which is racism. My how horrible they are! Poor, brave XYZ for standing up to them. I will vote for him'

I think it's well settled that the First Amendment applies to public colleges and universities. — Ciceronianus the White

Could you please elaborate on what that means? Does it just mean that the amendment protects a person from prosecution, or administrative sanction, for views expressed in a lecture, tutorial or more widely within public university grounds?

Or - a much stronger interpretation - that a public university is legally obliged not to refuse to hire out its facilities to someone on the grounds of the views they express, unless the expression of those views would actually break a law?

I would be surprised if the latter were the case but, being an alien, I never cease to be surprised by things I learn about God's favourite country.

I apologise unreservedly, wholeheartedly and ashamedly for mentioning the placard on the lectern of Jordan Peterson (remember him?). It seems to have derailed the thread for the last two pages, which was not my intention.

Sorry, I couldn't get through it. I love Canadians and Canadian accents, and I like that Peterson talks thoughtfully and intellectually rather than haranguing angrily, but he takes way too long to get anywhere (he rambles all over the place!), and I'm not going to sacrifice 1 hour and 12 minutes plus of my precious time to see if he can make any sort of a case for a proposition that sounds ludicrous to me - that Marx somehow is responsible for identity politics. I find Marx persuasive and I find identity politics very unhelpful and I'd be very surprised if there were any credible connection between the two.

If you have a version of the argument that can be put in a few paragraphs, I'd be happy to read it, but my time's too precious to waste on that video, delightful though his accent may be.

PS It doesn't help his case at all that the lectern he's at has a big sign saying 'TRUMP Hotels'. It's not his fault, just bad luck, but it reduces his chances of changing anybody's mind to just about zero.

For me it depends on the nature of the 'de-platforming'. I'm all in favour of the owners of platforms, such as lecture theatres, town halls, TV and radio stations, social media sites etc, declining to make their facilities available to racist speakers (or censoring racist material in the case of online media).

I am not in favour of mobs of students blocking entry to lecture halls that a university has, however unwisely, decided to hire out to a speaker that is deemed racist, or their trying to prevent the speaker from being heard. This is for tactical, not moral reasons. Images of aggressive students shouting somebody down or blocking people from peacefully attending a lecture are high-octane fuel to the populist narrative of the Trumps and Milo Yiannopoulos's of this world, and do enormous damage to progressive causes.

...is not a sentence because it lacks a verb.

From what little I've seen of Jordan Peterson, I don't object to much of what he says. But I think the phrase 'the lie of white privilege' is silly.

White privilege is simply not having to wonder whether a stranger will suddenly start to abuse you on the bus, just because of what you look like. In the US it is also not having to fear a police officer every time one comes near, that they may stop and search you, or even shoot you, because of what you look like. One would have to live under a rock to think that such a privilege does not exist.

Where claims about white privilege become silly is when they start to imply that ALL white people are better off than ALL non-whites, and there do seem to be plenty of extremists that say or imply such things. But the fact that the notion of white privilege may be misused by silly people does not imply that the notion itself is flawed.

My sensory faculties just aren't refined enough to detect them. Does the paper exist when I'm not observing the resonances? A further inference still seems to be needed.

This raises an interesting question about what we mean by 'detect', or 'perceive'. Specifically, do we want 'perceive' to mean the same thing as 'notice'?

My eardrums will be vibrating in a slightly different way from how they would if the paper were not there. And maybe even the electrical signals sent along nerves from my ear to my brain are slightly different. But I do not notice these differences. Would we then say that I do not perceive them, even though the difference in information is reaching my brain?

The famous psychology experiment about the gorilla on the basketball court provides a super example to focus on this question. The people's sensory organs detected the gorilla, and the signals about that reached the watchers' brains, but the watchers did not notice the gorilla. Would we then say that they did not 'detect' or 'perceive' the gorilla.

What if the gorilla and all the basketball players were robots (so that none of them can be conscious of the gorilla) and the only animals in the room were watchers, none of whom noticed the gorilla. Would we say the gorilla was 'unperceived' and if so would the question about whether the gorilla existed be essentially the same as for the paper in the drawer?

Yes, it was the same as in your latest post. What I meant was that it was different from the article you linked in introducing this sub-thread, which was about the diagonal lemma. But I don't think that matters, and I wonder if we have been talking past each other, while really agreeing all along. My reading of that latest article is that Tarski used the diagonal lemma to show that, if there is a 'True(...)' formula in a strongly self-referential language L then a contradiction must arise. It follows from that that such a language cannot have a 'True(...)' formula, so that the Liar sentence cannot be expressed in it. Hence, any string of symbols purporting to implement the Liar sentence must fail to do so, either syntactically (it is syntactically invalid) or semantically (it implements some other meaning).

In other words, Tarski proved that sentences that look like Liar sentences are really just meaningless confusions arising from a failure to be sufficiently formal.

Do you agree with that interpretation?

Tarski proved a stronger theorem than the one stated above, using an entirely syntactical method. The resulting theorem applies to any formal language with negation, and with sufficient capability for self-reference that the diagonal lemma holds. First-order arithmetic satisfies these preconditions, but the theorem applies to much more general formal systems. — Wiki

That quote is from a different article (this one), and what it refers to as 'the one stated above' is not the Diagonal Lemma.

The article goes on to say that Tarski's Undefinability Theorem - the theorem the article is about - says there cannot be a formula in the relevant language L that defines T*, the set of true formulas in (L,N). I interpret that as implying that the 'Liar sentence' is not expressible in a formal language, even one with 'sufficient self-reference', because there is no 'True(...)' formula or predicate.

At the end of the article it says

An interpreted language is strongly-semantically-self-representational exactly when the language contains predicates and function symbols defining all the semantic concepts specific to the language. Hence the required functions include the "semantic valuation function" mapping a formula A to its truth value ||A||, and the "semantic denotation function" mapping a term t to the object it denotes. Tarski's theorem then generalizes as follows:No sufficiently powerful language is strongly-semantically-self-representational. — wiki

I am interested in exploring this more. I do not have much familiarity with Tarski's work.

The paper in my drawer is not heard. — PossibleAaran

It is though, as I explained in this post (which was on page 2 of 9, so it's understandable that it has been forgotten).

The resonance of your footsteps on the floor, and even the micro-audible vibration of the desk as air moves over it, will differ according to how many pieces of paper are in the desk drawer.

In order to talk about things that may have no effect at all on your sensory organs, you need to at the minimum change the focus to objects outside your past light cone, which means objects in distant outer space. There are difficulties there as well, but they are different difficulties.

The T-schema for instance uses the diagonal lemma and so can produce the Liar. — MindForged

Are you sure it can do that validly? The linked page states the lemma with a premise that restricts it to first-order languages, which I expect would rule out its use in a T-schema environment which I believe is higher order.

I was going to check the proof to see if that premise is actually used, but I got tired and didn't, so I'm hoping maybe somebody else did. It would be unusual to state a premise that was not used though.

I think all you're really doing is denying the possibility of self-reference, because the Liar is constructed within a semantically closed language — MindForged

I had assumed we were discussing within that context, along with the inconsistencies and explosions that inevitably flow from that. Why do you think that involves denying the possibility of self-reference? My expansion of the sentence to the more formal version above is following how Russell expands 'The present king of France is bald' in his theory of definite descriptions, not seeking to forbid self-reference. The aim is to make explicit the implicit assertions hidden within a definite description.

The fact that my expanded sentence still contains the word 'this sentence' should be sufficient to demonstrate that the operation did not banish self-reference.

If you are being "excruciatingly literal minded" then you wouldn't substitute the truth value in for the referent of the sentence. The truth-value is part of the sentence that's being referred to, that's the Liar. — MindForged

I'm afraid I don't know what you are referring to with the words 'the Liar'. And also, I'm afraid I can't make anything of your first sentence. In my understanding, a sentence does not have a referent, it is names or symbols that have referents.

Hey, here's something really cool!

I just noticed that the sentence

'This sentence is True'

or more literally, per the post I just made:

'There exists x such that x is the truth value of this sentence, and x=True'

has two different solutions. It is satisfied not only by the hypothesis that x exists and is True, but also by the hypothesis that x exists and is False.

Try it and see!

That's made my day.

The liar isn't "False is false". — MindForged

The liar sentence, as usually given, is

'This sentence is false'

But if we are being excruciatingly literal-minded, it is a simple, false sentence, because a sentence is a bunch of words and 'false' is a truth value, so the two are not the same thing (a bunch of words is not a truth value), and should not be connected by the word 'is', which implies identity.

Hence we interpret it as meaning:

'there exists x that is the truth value of this sentence and x = False'

If the truth value of the sentence is 'true' then, under the axiom schema of substitution (see I9 from here), we can substitute 'true' for the words 'the truth value of this sentence', without changing the truth value of the sentence. That gives us

''True is the truth value of this sentence and True = False'

whose truth value is False, because the second conjunct is.

Certainly not self-reference because, as you mention, there are self-referential sentences which don't pose a problem. Something else about the liar paradox (in conjunction with self-reference) causes the problem, but it isn't a given that this "something else" is the same thing for both "this sentence is false" and "this sentence is meaningless" (or "this sentence is either false or meaningless."). — Michael

I agree. It is not the self-reference alone that is the problem.

My interpretation is that the problem is that the sentence refers to its own meaning, as assessing Truth requires first assessing meaning. So one has to work out what it means before one can work out what it means - there's the problem right there.

That's why sentences like 'This sentence is written in English' or 'This sentence has ten words' are not viciously circular. They are self-referential but the reference is to the sentence's syntax, not to its semantics (meaning). So one needs to only observe the sentence's syntax, not its semantics, before one works out its semantics.

I suppose it's that the sentence's truth value is referential that is the problem. Whereas in the examples of the last paragraph the truth value is not self-referential.

Self-referentiality need not be a problem. In mathematics, we hit self-referentiality when we have 'x', whose value we want to find, appearing more than once in an equation. Sometimes we can solve that by rearranging the equation so that 'x' appears only once, on one side. We call that 'making x the subject of the equation'. For example we can solve the equation 'x = 1/x' by re-arranging it to be 'x^2=1', so that x = sqrt(1), with solutions x=1 or x=-1.

Sometimes we can't do that but we can still guess solutions or find them by numerical analysis, as with equations like 'x = sin x'. What that involves is basically guessing a solution, seeing how it goes, and then refining it if necessary.

That approach works with the sentence 'This sentence is True' (quoted by ). We can't solve it deductively because of the vicious circle on truth value, but we can guess that the answer (solution) might be 'True' and when we substitute that for 'This sentence' in the sentence, we find that we get 'True is true', which works, and which is consistent with our tentative hypothesis (guess) that the sentence is true.

But it doesn't work with 'This sentence is false' because if we guess 'false' and substitute, we get the sentence 'False is false', which is true, which contradicts our tentative hypothesis that the sentence is false.
And if we guess 'true' and substitute, we get 'true is false', which is false, which contradicts our tentative hypothesis that the sentence is true.

So we can't guess a solution either.

My analogy is that deductively working out the truth value of a sentence, by interpreting it and assessing its truth value, is analogous to making 'x' the subject of the equation and working out the value of the other side. We can do that for 'That pot is black', 'This sentence has ten words' and for 'x=1/x', but not for 'This sentence is true', 'This sentence is false' or 'x=sin x'.
And guessing a truth value and testing how it works is analogous to guessing a value of 'x' and testing how it works. We can do that for 'x = sin x' and for 'This sentence is True' and find a truth value / x value that works, but we can't do it for 'This sentence is false' because none of the values we can guess work.