Don't worry mate, you'll never hear from me again. — Wayfarer

LOL. I'm happy to hear from people who take the time to understand what I'm saying. Someone responded to my point about Judith Miller by claiming that I "don't know how Google works" because I noted that a search of "Judith Miller lies" returns over 17 million hits. My God. Anyone who thinks Judith Miller was sincerely mistaken rather than a deliberate war propagandist is either a fool or a neocon maniac, two species of human in great supply these days. I need all the understanding I can get around here.

Well, fair enough! That's why I asked. — Wayfarer

Thanks, I added a couple of more paragraphs. I'm a liberal in despair at what's become of liberalism. Those two debates this week have got me in an awful mood. I'm for Tulsi, who's for peace. You see she was the only one to get a hostile question. Why is that? The left has abandoned peace and craves war with Russia. Don't get me started.

So it seems to me that your leading with a criticism of the New York Times is an attempt to divert the thread away from the topic of the relationship of 'conservative' politics and media, by trying to prove that what is generally called 'the liberal media' beat Fox and the other "conservative" media outlets to it. Would that be right, or am I misreading you? — Wayfarer

Deeply misreading me. My point is that fake news is used these days to label what I would call alternative news, any questioning of the mainstream narrative. I used the WMD example to make the point that although the NYT often gets the story right, when they get it wrong the consequences are awful. I can watch a hundred flat earth videos on Youtube and they do no damage. They're harmless. When the esteemed NYT publishes the fabrications of Judith Miller and we're still in Iraq 17 years later, that supports my point that by the metric of REACH and INFLUENCE, the NYT is the greatest purveyor of fake news in the world.

But virtually everything I post here is misread. When I attack the left, I am coming from the left. The people who keep me sane are Glenn Greenwald and Jimmy Dore; liberals who detest what's become of the liberals lately.

After all, why would I be upset about Hillary Clinton (and Joe Biden, who's been in the news lately ...) for their votes on the Iraq war if I were myself a neocon maniac lusting for war? It was Hillary's vote that gave cover to the rest of the Democrats to vote for the war; and that was for me one of the final straws in my break with what's become of liberalism these days. The reign of Obama did the rest, when he institutionalized the wars and the torture. That's why I really blew up a couple of weeks ago when someone said that "Trump puts kids in cages." Obama's record on Mexican immigration was atrocious. He deported record numbers of Mexicans, far far more than Trump has (in equal amounts of time). He put kids in cages. He turned kids over to traffickers, which has better optics than separating alleged families long enough to determine if they really are a family.

So when I'm angry about these things I look like a Trump supporter. I'm not. I'm a liberal in total despair at what's become of liberals. Did you see that debate last night? "Trump put kids in cages," "Trump KIDNAPPED kids." Good God. Trump is a LIGHTWEIGHT compared to what Obama did. I'm not in favor of Trump's awful immigration policy. I'm simply in despair at the childishness and willful ignorance of what passes for leftist critique of immigration policy.

The law of identity operates at a much "lower level" than that of modeling changing systems like weather or biology.
— fishfry

Yep. Or basically what we talk is about a bijection. Or set theory. — ssu

Identity is deeper than bijection. There's a bijection but not identity between {1,2,3} and {a, b, c,}.

The term "fake news" is overwhelmingly associated with the election of Donald J Trump, who popularised the term by smearing the media on every available opportunity and saying every criticism of him was 'fake news' — Wayfarer

I remember it the other way 'round. The left started the term fake news with that bogus PropOrNot article in the Washington Post, which had to be retracted after even fair-minded liberals saw how unsourced and fake it was. Trump appropriated the term for his own use. But we can retcon the phrase to famous historical incidents like the attack on the Lusitania (loaded with illegal munitions hence fair game for the Germans), the Reichstag fire, the Gulf of Tonkin attack that never happened, the WMDs, etc. Fake news is as old as the Trojan horse. Just ask Goebbels. Or for that matter Edward Bernays, the great theorist of propaganda as a tool for governments to lead the people. "His best-known campaigns include a 1929 effort to promote female smoking by branding cigarettes as feminist "Torches of Freedom" and his work for the United Fruit Company connected with the CIA-orchestrated overthrow of the democratically elected Guatemalan government in 1954 ..."

https://en.wikipedia.org/wiki/Edward_Bernays

OMG my mentions are really piling up. What actually happens is that every time I write anything on the politics forums the replies depress me terribly and I have to stay away for days. I should know better. Back to math.

First of all, the point about formal systems. I completely agree that the formal proof is not the essential part of a theorem about geometry! I'll tell you more: I am pretty convinced that the formal proof alone does not contain the essential information necessary to "understand" the theorem, and I started to write on this forum because I was looking for somebody that has some results/ideas on this point. Please take a look at my last post on the discussion that I opened about two months ago: "Is it possible to define a measure how 'interesting' is a theorem?" (https://thephilosophyforum.com/discussion/5789/is-it-possible-to-define-a-measure-how-interesting-is-a-theorem).
Then, since nobody seemed to find the subject interesting, I started to reply to posts about infinity.. :-) — Mephist

I didn't read that thread because I thought the answer was trivial. You can measure citations of a paper, and that is commonly done. It works the same way as Google's page rank algorithm. Alternatively, it's unanswerable, because sometimes it takes decades or even centuries for the value of a mathematical insight to become apparent. But I'll glance at that thread if you like, once I catch up with my mentions. Which never happens.

But the point of the discussion about infinity was more or less this one: "are ZFC axioms about infinity right?" and my answer was: "you cannot use a formal logic system to decide which kind of infinity exists". And I said you that I "don't like" ZFC because, for example, of Banach-Tarski paradox. — Mephist

Well of course no axiomatic foundation for math can be right. People did math for thousands of years without any foundation at all, and now we've got several. But I would draw a line in the sand here and say that set theory has been extraordinarily successful as a theory of infinity, and I don't know that HOTT or Category theory are any improvement at all.

Of course no foundation can be "right" about infinity until the physicists discover actual infinity in the universe and report back to us. Just as we didn't know whether Euclidean or non-Euclidean geometry was right until Einstein's great breakthrough.

Regarding B-T, surely it's not right about the world (unless it is -- work for the physicists again). But as a theorem of ZFC, it's as right as a legal position in chess. Which is to say that it's the endpoint of a legal sequence of moves in a formal game.

Let me try to explain this point: I am not saying that ZFC is wrong and Type Theory, or Coq, is right. — Mephist

That's good, because the claim is meaningless. There is no right or wrong about it. Only useful or not, beautiful or not, interesting or not, fashionable this century as opposed to last.

I am saying that the encoding of segments (and even surfaces and solids) as sets of points is not "natural", — Mephist

There was a poster here a while back who gave me some great insights along these lines, before becoming so insufferably rude that I could no longer interact with him. I learned that Charles Sanders Peirce noted that the set-theoretic continuum can't possibly be the right model of a continuum, because a continuum must have the property that every part of it is the same as the whole; and a set, being decomposable into the union of its singleton points, fails this test.

If that's what you mean, I take your point. But knowing what's the "right" model of the continuum is beyond my own personal pay grade. I know that Weyl and Brouwer and others had many deep thoughts along these lines. I know little of these matters.

What I do know is that the set-theoretic continuum, having won the 20th century, underpins all modern physical theory. If one seeks to overthrow that foundation, one has much work to do.

because you can build functions that have as input a set of definite measure and as output a set where measure is not definable: the transformations on sets and on measures are not part of the same category! Of course, you can say that this is because there really exist geometric objects that are not measurable, and it is really a feature of geometry, and not of ZFC. — Mephist

If we accept the point-set foundation, then it's reasonable that not every point-set is measurable. After all, there are some wild point-sets. If you reject point-sets, then I guess not. But nonmeasurable sets in ZFC depend on the axiom of choice, and perhaps that's what you object to. It's AC that gives us sets whose only property is existence, with no clue as to how to construct them or identify their elements.

Well, in that case you have to admit that there exist several different geometries, — Mephist

Well yeah, but the Euclidean/non-Euclidean example gives us a good historical datapoint. There are lots of geometries but only one "true geometry" of the universe. But I'm not interested much in applications. The physicists can do their thing. I'll wear my formalist hat and steer cleer of ontological issues.

because there are sound logic systems based on type theory where all sets are measurable and lines are made of a countable set of "elementary" lines (integrals are always defined as series) (elementary lines have the property that their squared length is zero), and Banach-Tarski is false. — Mephist

I'll take your word for that. Are you talking about SIA? What of it? There are variations of chess too. It doesn't bother me.

But why are you hung up on Banach-Tarski? If you don't like point-sets, your objecions to standard math would be much wider than that.

So, we can ask which one of the possible geometries is the "right" one. — Mephist

As meaningless as the question of which variant of chess is the right one. Or whether in some fan fiction, Ahab is the cabin boy and not the captain of the Pequod, and whether that's right or wrong.

What do you mean by right one? Right as in best foundation for math? Or right as in true about the actual world? I don't share your focus on trying to figure out what's right about things that you and I will never be able to know, unless the next Einstein hurries up and gets born.

The problem is that they are only mathematical models. From my point of view, this is exactly the same thing as mathematical models for physics (or maybe you want to consider euclidean geometry as a model, but in that case non measurable sets surely do not exist): the best mathematical model is the one that encodes the greatest number of physical results using the smallest number of physical laws, and none of them is perfectly corresponding to physical reality anyway. — Mephist

That's a pragmatic requirement I don't share. You think math is required to be bound by physics. Riemann kicked the hell out of that belief in the 1840's. You think math is physics. Mathematicians don't share that opinion. Math is whatever mathematicans find interesting. If the rest of the world finds an application, more the better, but that is never the point of research in pure mathematics.

I was looking for a formal proof of BT in ZF Set theory — Mephist

There could never be such. ZFC is necessary.

because the critical part of BT is the function that builds the non measurable sets (the one defined using of the axiom of choice). — Mephist

Right. So why look for a proof in ZF that could never exist?

You are right that in most cases the formal logic is not necessary at all, but in this case the result of the theorem depends in a critical way on the actual encoding of a continuous space as a set of points. — Mephist

I really don't follow your point here at all. B-T is the least of it. If you reject point-sets as the foundation of math, that's fine, but then you have to rebuild a LOT of stuff. B-T is the very least of it. I don't understand why you regard B-T as uniquely interesting in the context of completely re-founding math on non point-set principles.

In fact, I am pretty convinced that It's possible to encode euclidean geometry in ZF Set theory in a way that is perfectly sensible for all results of euclidean geometry but makes BT false (I think I can do it in Coq, but surely you wouldn't like it.. :-) ) — Mephist

Why wouldn't I like it? Why would I even care? You seem to be tilting at windmills that I'm not defending. All I've ever said about set theory is that it won the 20th century and that Planck noted that science progresses one funeral at a time. Surely that shows my open-mindedness about historically contingent trends in foundations.

And as I already pointed out, if set theory is no longer required to be foundational, it becomes interesting for its own sake. It makes no difference to practitioners of set theory whether it's foundational or not.

That is not true: the definition of measure is purely algebraic, and can be used with ANY definition of sets: it's the same thing as for calculus, derivatives, integrals, etc.. — Mephist

I don't know what you mean but reading ahead I think I do so let's defer this for a moment ...

Well, I think you can easily guess: I studied electronic engineering (Coq can be used to proof the correctness of digital circuits), but I am working as a software developer. But I am even interested in mathematics and physics, and even philosophy, so I think I am not the "standard" kind of software developer.. :-) — Mephist

Well, engineers definitely have a pragmatic view of math. And one of my ongoing theses is that there are a lot of people studying computer science these days who thirty years ago would have studied math. So the mathematical point of view is somewhat lost among many people these days. But when you said that the purpose of a math foundation is to be able to found physics, that's just wrong. It's not how mathematicians view it at all.

I didn't say that I don't like point sets: I said that point sets are not a good model for 3-dimensional geometric objects, — Mephist

Ok. I can't argue the point but even if I could I don't have much reason to.

and I am not very original with this idea: in HOTT segments are not sets but 1-spaces. The word "sets" is defined as synonymous of 0-space (or "discrete" space). — Mephist

Ok. I wasn't aware that HOTT had gotten to the point of replacing differential geometry in general relativity, nor functional analysis in quantum physics. Are you making those claims? If so I am not sure I believe you. I watch a lot of physics videos these days and they're all differential geometry and functional analysis. The Schrödinger equation is functional analysis, not homotopy type theory. I think you're making claims that are more aspirational than true of the current state of the art.

Point-set topology and algebraic topology are equivalent, and all modern mathematics and physics since Grothendieck (https://en.wikipedia.org/wiki/Alexander_Grothendieck) are based on category theory: as you just said, they really don't care much about foundations... :-) — Mephist

Well yes and no. Analysis hasn't been categorized much. But physics? Again I think this is an aspirational claim. Loop quantum gravity uses n-categories but as far as I know relativity and quantum theory use classical 20th century math.

Both string theory and loop quantum gravity (the two most trendy at the moment....) (https://en.wikipedia.org/wiki/Loop_quantum_gravity) make heavy use of algebraic topology: not only they don't care "of what are made" the objects of the theory, but they are even not clear if they are something geometric, or maybe can be interpreted as emergent structures coming from something even more fundamental. This is the point of view of category theory: don't describe how objects are made, but only how they relate with each-other. — Mephist

Yes ok but LQG is not the standard theory yet. You're being aspirational again.

I was speaking about changing the topology of a set: any open set (that has no minimum or maximum) can be transformed into a set with a minimum (lower bound), not open and not closed. — Mephist

You are misapplying the well-ordering theorem. Bijections can radically change the order properties of a set. And the well-ordering theorme is equivalen to the axiom of choice so what of it? Reject one, reject the other. Take it up with Zermelo.

If the isometries used in BT are continuous transformations, they should preserve the topology of the sets that are transformed. — Mephist

Isometries preserve measure. They're rigid motions in space.

My guess is that the ones used in BT cannot be continuous on sets that are not discrete (that is one of the things that I wanted to understand from the formal proof..). — Mephist

I'm not sure exactly what you mean by continuous on sets that aren't discrete. All isometries are continuous and measure preserving (on sets that have a defined measure), period.

In other words, I think that to make that transformation you have to destroy the topological structure of the object (but I am actually not sure about this). — Mephist

I don't think so. Isometries are continuous since they preserve distances, and they preserve measure.

If this wasn't possible (and, as I understand, it's not possible without axiom of choice) I think BT would be false. — Mephist

Isometries aren't mysterious, they're rigid motions of space. Flips, rotations, translations. Clearly continuous, measure preserving, "shape-preserving" if you like. If you translate a triangle it's still a triangle. This is not mysterious nor does it require any esoteric philosophical assumptions.

— Mephist

Yes, it's even more interesting because of the fact that it is an (apparent) paradox, and paradoxes are the most informative and interesting parts of mathematics. — Mephist

It's a veridical paradox, meaning that it is NOT actually a contradiction, it merely violates our intuition. It should be named the Banach-Tarski theorem, since that's what it is. Now if you want to argue from that that the axiom of choice should be rejected, or points should be rejected, or sets should be rejected, that's fine. But it doesn't alter the fact that B-T is a theorem of ZFC, and moreover, not an extremely difficult one.

And as I mentioned earlier, it rests on the paradoxical decomposition of the free group on two letters, which at some point I should talk about, because this is a very strange paradox that has nothing at all to do with points or the axiom of choice or anything else questionable. It's very very simple and natural.

Well, I have my doubts here. These are rigid motions on countable subsets of points, it means only on subsets with zero measure. — Mephist

No not so. An isometry is a rigid motion on countable and uncountable sets. If I take the unit square and translate it 3 units to the right and 3 units up, the shape and measure are preserved. All measures are preserved.

I think you should open up a separate thread on Banach-Tarski so we could talk about these issues without all the philosophical baggage. You are misunderstanding a lot of basic issues. If you have a point set of any cardinality and measure, an isometry preserves its cardinality and measure. And shape, even though that's not a technical definition.

I am quite sure that they cannot be continuous transformations on measurable subsets with non-zero measure. — Mephist

You're simply wrong on this technical point. You should open a separate thread on B-T because we can't walk through that proof while you're claiming physics is based on HOTT and that mathematicians have point sets wrong. These are very different issues.

Regarding B-T you are misinformed on the basic math, on issues that are very clear and simple, and that we should discuss separately.

If you have a proof that they are, I am very interested in it ( even not formal :-) ) — Mephist

That a reflection, rotation, and translation are continuous? A freshman calculus student could work out those proofs. Measure preserving? A little trickier but not difficult.

Come on. Are you doubting that a translation is continuous? A reflection? A rotation? Why are you making these claims whose disproof is so elementary? I don't follow your logic here. Don't mean to be piling on, but after all this heavy-duty category theory and philosophy you are missing some very elementary freshman calculus.

You have a set of points which you can think of as vectors in the plane or space. You add a constant vector to each point. The resulting point set has the same shape and measure, and the transformation is continuous. I can not accept that this is not obvious to an engineering major.

P.S. Algebraic definition of measure: see https://en.wikipedia.org/wiki/Sigma-algebra — Mephist

Yes ok what of it? From the basic axioms of measure one proves the existence of a nonmeasurable set. And from that, Banach-Tarski.

I gather you are trying to make the claim that measure theory is "algebraic." That's a point of semantics. From countable additivity plus the axiom of choice, the existence of a nonmeasurable set falls out. Have you seen the proof? It's not difficult. Well it's not difficult after one's been through it a few times. Like everything else. But it's a very cool proof. It's a paradigm of all axiom of choice proofs.

But the bottom line, and this has been a long post so here is the tl;dr:

Your interest in Banach-Tarski is orthogonal to all the philosophical and foundational issues; and the confusion generated by conflating these things is causing you to misunderstand some extremely elementary points of math, such as the continuity of translations and rotations and reflections.

Ok one post down, so many more to go ... thanks for reading.

ps -- Here is the proof that an isometry preserve a Hausdorff measure, which the measure on Euclidean 3-space is an example of. In the general case it's not true but that case doesn't concern us. https://math.stackexchange.com/questions/695492/isometry-vs-measure-preserving

pps -- Simpler theorem on point. Isometries preserve Lebesgue measure of Euclidean space.

https://math.stackexchange.com/questions/242837/lebesgue-measure-is-invariant-under-isometry

Are you claiming that Judith Miller was guilty of deliberate fabrications? What evidence do you have of this? — Fooloso4

I Googled "Judith Miller lies. Iinterestingly, when I Googled "Judith Miller," Google autocompleted "lies" as the first suggestion.

I was going to start posting links, but to save wear and tear on my fingers you can click for yourself. There are over 17 million results.

https://www.google.com/search?ei=FsQTXf-MG-iU0gK2moigBw&q=judith+miller+lies&oq=judith+miller+lies&gs_l=psy-ab.3...0.0..22328...0.0..0.0.0.......0......gws-wiz.Glrx2suS-Sw


That anyone would respond to my post by claiming Judith Miller had no agenda when she used the NYT to help lie the country into war is ludicrous. The facts are well-known. You still hanging on to hope the WMDs will be found? There were no WMDs. The NYT helped Bush lie the country into war. If you don't know this, you're the last person in the country to find out.

[ THIS WAS THE LAST PART :-) ] — Mephist

Ok! In the meantime I just read through your first reply and I believe I can respond concisely and beneficially to it. Let me get to that. It might take me a day, or basically five minutes after putting it off for a day. But when I wait a day the five minutes worth is generally better. So let me work on reply #1 and read through the others.

I do think it's very cool that you learned Coq in EE. I had no idea! My background is that I learned set theory-based math back in the day, but my education did include introductory category theory as part of grad level algebra. In fact it is one of the high-water marks of my mathematical career that I totally grok the definition of the tensor product of modules as a universal property. I get a lot of mileage out of that one example!

I spent my professional career as a working software engineer and programmer. Along the way I was on Usenet in 1995 and started reading John Baez's amazing This Week's Finds in Mathematical Physics, which was essentially the world's first math blog. I was astonished to see that "n-categories" were being used to do loop quantum gravity. When I was in school categories were only for mathematicians, there were no applications yet. Now it's all over physics and computer science and biology and who knows what else.

I've followed all this at least via Baez over the years, and now that I've recently been reviewing and learning some more category theory, I am starting to get a vague understanding of what an n-category might be. And along the way of course I watched Vovoedsky's videos and vaguely kept up with HOTT at least at a buzzword level. It helps a lot that I already knew what a homotopy is from algebraic topology back in the day. So I get the thing about paths and paths between paths.

I think we have a good meeting of backgrounds coming from totally different directions. Now I'll go see what I can do with the reply queue.

OMG am I three posts behind already? And you know me, I sometimes reply at length to every paragraph. I'm overwhelmed again. Just remembering back, my favorite thing that I said was about the ordinals. The ordinal numbers are very cool, so I hope you said something about them. I'm going to go read at least your first reply and maybe the other two and mull them over a while.

Definitional question. In 2002 the NYT ran stories by Judith Miller alleging that Saddam Hussein was acquiring yellowcake uranium and aluminum tubes for the construction of WMDs Those articles, appearing as they did day after day after day in the Paper of Record (TM), helped turn the tide of public opinion in favor of invading Iraq, and gave cover to politicians (Hillary and Joe Biden to name two) to vote for Bush's war despite millions of liberals (who used to be against war, way back in the day) marching in the streets against it. That was by the way the last time we saw anything from the anti-war movement in this country. Something that troubles me.

As it turned out in the fullness of time, those articles were lies. To be absolutely clear, they were not well-intended mistakes. They were deliberate fabrications for the purpose of lying the country into war.

Now, would you or would you not define that as Fake News?

If you say yes, then in terms of reach and influence and bloody consequences, the NYT is the greatest purveyor of Fake News in the world.

On the other hand if you say no, that BY DEFINITION whatever is in the Times isn't fake news because the definition of Fake News is NEVER what the Times prints, it's only what people QUESTIONING the Times print. So it's more of a definitional thing, having nothing to do with whether a consequential news story happens to be true in any objective sense. If the Times prints it it's not Fake News.

So pick one. NYT stories on Saddam's WMDs that drove the country into a disastrous war that we're still stuck in: Fake News or not Fake News?

Blaise Pascal (1623-1662), in his Pensees, writes: "All of humanity's problems stem from man's inability to sit quietly in a room alone." (I would add, DOING NOTHING, which is how I'm sure he would have intended the statement to be understood) — miguel d

Later in life his wife took up with another man, leading the neighbors to gossip about Pascal's triangle.

OK, I read the article and finally understood the point about isometry group! :smile: — Mephist

Ok! And I'm so glad you responded. I come to this forum to discuss math, and when there's no interesting math content going on I go over and get in trouble on the political forums. Discussing math is much more productive. As I saw on a mathematician's office door once: Good sense about trivialities is better than nonsense about things that matter!

Actually, I tried to look for a formal proof of Banach-Tarski in ZFC Set theory, but the only one I found is using Coq... :-) (https://hal.archives-ouvertes.fr/hal-01673378/document) — Mephist

Arghhh. In my opinion you are totally missing the point of math. I don't mean that to be such a strong statement, but in this instance ... yes.

The point isn't to have a pristine logical proof. The point is the beautiful lifting, via the axiom of choice, from the very commonplace paradoxical composition of the free group on two letters, up to a paradoxical composition of three-space itself. It's the idea that's important, not the formal proof. That is the actual point of view of working mathematicians.

I recently ran across an article, link later if I can find it. Professional number theorists were asked by professinal logicians, wouldn't you be interested in seeing a computerized formal version of Wiles's proof of Fermat's last theorem? And the logicians were stunned to discover that the mathematicians had no interest in such a thing!

What's important about Wiles's proof is the ideas; not every last bit and byte of formal correctness.

This is something a lot of people don't get about math. It's the overarching ideas that people are researching. Sure, you can work out a formal proof if you like, but that's more like grunt work. The researchers are not interested.

Likewise, you are interested in a formal computer proof of BT, and that is not the point at all. The point is that the paradoxical decomposition of the free group on two letters induces a paradoxical decomposition of three space. That's the point. It's beautiful and strange. Formal proof in Coq? Ok, whatever floats your boat. But that is not the meaning of the theorem. The meaning is in the idea.

I hope I'm explaining the modern viewpoint here. The ideas are important, the proofs are not. I know, that's the opposite of what they tell people in the philosophy of math classes. But it's how mathematicians think.

I found the article.

Why the Proof of Fermat’s Last Theorem Doesn’t Need to Be Enhanced

Decades after the landmark proof of Fermat’s Last Theorem, ideas abound for how to make it even more reliable. But such efforts reflect a deep misunderstanding of what makes the proof so important.

It's a great read, it makes my point much better than I can.

https://www.quantamagazine.org/why-the-proof-of-fermats-last-theorem-doesnt-need-to-be-enhanced-20190603/

- By the way, there is a formal logic computer system using ZFC Set theory with a very extensive library (http://mizar.org/), but there is no Banach-Tarski theorem there (at least at the moment) — Mephist

That may well be true, but it is in no way relevant to the interestingness of the theorem!

I am looking for formalized proofs because usually proofs in ZFC related real numbers are too long to be completely written by hand, and tend to be not clear on the steps that make concrete use of ZFC axioms. — Mephist

That's perfectly cool if you are interested in detailed step-by-step proofs derived from first principles. But that's a different interest than the math itself.

Anyway, the reason of my doubts on ZFC in geometry are more "fundamental", and not related to the existence of the isometry group. — Mephist

Two points here, one, I didn't realize you have "doubts" about ZFC, but if you do I might try to alleviate them or even perhaps agree with them. "ZFC in geometry" I didn't follow, you mean modern geometry ie algebraic geometry (very category theoretic), geometric algebra, etc.? Not catching your reference.

But secondly, you spoke of the "existence of the isometry group" as something strange or questionable. Perhaps you mean something else. If you take Euclidean 3-space straight out of multivariable calculus class, it's clear that if you have an object or set of points, you can transform them in a distance-preserving manner; and that the collection of all such transformations forms a group. Surely nothing is radical or counterintuitive about that.

If you don't like the idea of "space as a set of points" I do understand that philosophical objection. But then you are objecting to a huge amount of modern math and physical science too. I'm not sure where you're coming from. If you don't like point sets, then you wouldn't like set theory. I can certainly see that.

The thing that I don't like is that in BT (as in all topology based on ZFC), a segment (or a surface, or a solid) is DEFINED to be a set of points. — Mephist

Yes that is certainly the case. And I get that you are objecting to that perspective. Which is fine. If you are engaged in the century-long project of replacing set theory, which won the 20th century; with HOTT (or category theory or Smooth Infinitesimal Analysis, etc)., which will probably win the 21st; I can't argue the point, we are all just passing through history.

But surely your objection then is not to BT, but to virtually all of modern mathematics and physics. Is that your viewpoint? How far does your rejection of using point sets to model mathematical ideas go?

(well since in ZFC everything is defined as a set, you don't really have much choice with this definition...) — Mephist

Even if set theory were dethroned as the foundation of mathematics -- and of course it already has been dethroned in much of modern math -- it would still be of interest. Just as group theory is the study of all the mathematical structures that satisfy the axioms for groups; set theory is becoming the study of all the mathematical objects that satisfy the axioms for sets.

In other words now that set theory is "relieved of its ontological burden," as I saw it expressed once, we will now see a great flourishing of set theory, not its demise. Set theory is fascinating whether you regard it as foundational or not!

So if you deny set theory as a foundation, I will not disagree with you! Rather, I will point out that we can now study set theory for its own sake. Do you object?

The problem is that you can't use the property of a segment being a set in any way, because equivalence relations on sets (one-to-one functions) do not preserve any of the essential properties of the segment:
- the size of the set is not preserved, — Mephist

Correct, the intervals [0,1] and [0,2] have the same cardinality but different measures. But so what? A quart of water weighs differently than a quart of oil. You can have two systems of measurement that are independent of each other. So what? When we care about cardinality we care about cardinality. When we care about measure we care about measure.

because additivity doesn't work for uncountable sets — Mephist

Sure, that's the age-old problem of the continuum. How can a big pile of dimensionless points give rise to dimension? Uncountable additivity would lead to a contradiction because the measure of the union of singleton points would be 0 but the measure of the unit interval is 1. So we can't have uncountable additivity and nobody knows how points make up a line. You might as well complain to Euclid.

(and I believe there is no way to define a sum of an uncountable set of terms that gives a finite result, — Mephist

It's a theorem that if you have an uncountable sum that converges to a finite number, all but countably many of the terms must be zero. Not that much of a surprise.

because the elements of an uncountable set are not "identifiable" with a computable function, and then you can't "use" them to compute a sum). — Mephist

I don't think that has anything to do with it. It's not about the inability to compute a sum. It's a simple proof but it doesn't involve computability.

- the topology of the set is not preserved, since every set can be reordered to have a least elenent (https://en.wikipedia.org/wiki/Well-ordering_theorem). — Mephist

The w.o. theorem does not come into play here at all. It's an interesting theorem though, worth a separate thread but a little off topic here. It's logically equivalent to the axiom of choice anyway so complaints about one are complaints about the other.

By the way you don't need the power of the well-ordering theorem to change the order type of a set via bijection. Just take the natural numbers 0, 1, 2, 3, ... and reorder them to 1, 2, 3, ..., 0. So it's the usual order but n '<' 0 for every nonzero n, where '<' is the new "funny" order relation. The natural numbers in their usual order have no greatest element; but in the funny order they do. We've changed the order type with a simple bijection. You don't argue that this simple example should be banished from decent mathematics, do you? The funny order is the ordinal number $\omega + 1$, or omega plus one. Turing knew all about ordinals, he wrote his Ph.D. thesis on ordinal models of computation. Even constructivists believe in ordinals.

I would like to call something out here. In your objecting to bijections, you are seemingly objecting to the operation of changing the order of an ordered collection. I wonder if you are going to far. Of course we can rearrange things. With infinite collections (don't call them sets if you like) we can change their order properties by rearranging them. Are you objecting to this as a concept or idea? What would be left in mathematics if we threw out rearranging things?

For this reason, the use of bijective functions with the "meaning" of moving a 3-dimensional object by breaking it into pieces and then reassembling it in another place is wrong. — Mephist

It's can't be right or wrong any more than the game of chess can be right or wrong. BT is undeniably a theorem of ZFC. That's true even if you utterly reject ZFC. The novel Moby Dick is fiction, yet it is "true" within the novel that Ahab is the captain of the Pequod and not the cabin boy. Banach-Tarski is a valid derivation in ZFC regardless of whether you like ZFC as your math foundation.

Suppose I stipulate that ZFC is banned as the foundation of math.

Ok. Then Banach-Tarski is still a valid theorem of ZFC. So what is your objection to that? B-T is still fascinating and strange and its proof is surprisingly simple. One could enjoy it on its own terms without "believing" in it, whatever that means. I don't think the game of chess is "true," only that it's interesting and fun. Likewise ZFC.

I mean: it's not contradictory, but it doesn't reflect the intuitive meaning that you give to the operation, since neither topology nor measure are invariant under this transformation. — Mephist

AHA! Yes you are right. But these are RIGID MOTIONS. That is the point. We are not applying topological or continuous transformations in general, which can of course distort an object.

We are applying RIGID MOTIONS. Isometries:

* Rotation around a point;

* Translation in a direction;

* Reflection through a plane.

These are rigid motions. They preserve measure. That is where you get the counterintuitive power of the theorem.

Let me say this again. We are NOT applying general bijections or topological transformations. We are applying only rigid motions in three-space, which are perfectly intuitive and reasonable. If you move an object from one location to another, or rotate it around a point, or reflect it in a plane, you preserve its measure if it has a measure. If not, then there is nothing to preserve.

The "trick" of breaking the transformation in two parts, so that the second part is only the composition of four pieces doesn't change the fact that the first part of the transformation does not preserve measure (the measure of each of the four pieces is undefined). — Mephist

Well yes. Distance-preserving transformations (isometries) have the following properties:

* They preserve the measure of measurable sets; and
* They map nonmeasurable sets to nonmeasurable sets.

So clearly the concept of nonmeasurable set is in play. And these are a consequence of the axiom of choice. HOWEVER! If you reject AC and thereby banish nonmeasurable sets, you lose all of modern probability theory, which takes with it a big chunk of modern physics, not to mention the social sciences. People didn't get in a room and say, "Let's foist nonmeasurable sets in the world because we're evil." They did it to get formal probability theory off the ground. You'll have to take up the foundations of modern probability theory with Andrey Kolmogorov.

[TO BE CONTINUED] — Mephist

I hope you won't get so far ahead of me that I despair of keeping up!

To focus the discussion, are you objecting to point-sets and ZFC in general? Then you object to a lot more than just B-T. But if you are interested in B-T for its own sake as a theorem of ZFC, we can talk more about that.

You have your figures and estimates all wrong. And that's just two billionaires. — Frotunes

This is something from a while back, I no longer remember exactly what I might have said and I didn't manage to find my earlier quote. I didn't follow your chart regardless. It shows discretionary spending of $1.1T. That doesn't include the other $2.7T, going by the 2015 total spending of $3.8T. Nondiscretionary spending includes all the social programs mandated by law., Social Security and Medicare and so forth, plus the interest on the debt (side question: Why do you think the Fed is doing everything it can for the past ten years to keep interest rates extremely low? Could it have anything to do with the disastrous effect on the federal budget if interest rates were allowed to rise in a free market?)

So even if you're trying to demonstrate that stripping billionaires naked would run the government for some nontrivial amount of time; showing a chart that represents less than 1/3 of government spending doesn't make your point. And the problem is that even if you could fund the government for a year or two by stripping the billionaires, what would you do in year three? There wouldn't be any billionaires. So you'd have to come after the millionaires, And after a couple more years, you'd be into the middle class. The middle class pays virtually ALL taxes no matter WHAT you do. The rich write the laws and the poor don't have any money. That leaves you and me to pay the bills.

Have you forgotten or are you just ignoring the fact that Trump brought the world to this precipice by backing out of an international agreement and putting a stranglehold on Iran with his sanctions? — Fooloso4

I'm perfectly well aware. I said the other day I favored Obama's Iran treaty. In my opinion a bad nuke deal is better than no nuke deal. Trump was 100% wrong on that, not to mention that showing the world that the US's word is only as good as the next election was simply a terrible thing to do.

I have a point of view that doesn't fit into the current paradigm of two warring sides that each regard the other as not only wrong, but uniquely evil.

My position, for the record, is that I agree with everything the left says about the right. And with everything the right says about the left. Our nation is in a very precarious position right now with few if any adults in the room. Jerry Nadler? That's your idea of a statesman?

So yes, Trump created this mess and now he's solving the mess he created. Yes I am perfectly well aware of that.

Thing is, I haven't heard much about Iran from the Democrats, in particular their presidential candidates. Cory Booker demanded that Biden apologize for consorting with segregationist southern Democrats (ie all of them of that era), and Biden said uh-uh no way am I going to apologize. On matters of war and peace? Crickets. If I missed one of the Dems saying something sane about foreign policy this week, I would thank you for a reference. I assume you are all familiar with Joe Biden's record on matters of war and peace. He's always been for war.

But that is Trump's style. Blow things up then calm things down. Just like his announcement that he was going to initiate nationwide immigration raids. Got everyone hysterical, then he cancelled that order too. That's his style. Without defending it in the least -- it's one of the worst things about him -- can't the left step back and stop getting triggered and going insane with anger every time he plays the same game? When will the left wise up?

Trump thrives on chaos. That doesn't mean YOU need to.

For the record, (1) My vote doesn't count, because I live in California, which will go strongly for any Dem candidate. But (2) to the extent that I can at least have an opinion, if not an actual vote, I would love to be able to vote for a Dem candidate. I don't see any besides Tulsi. Even Marianne Williamson is better that the rest of this crowd.

Bill Maher came out for Oprah. I am not alone in my low estimation of the crop of currently announced Democratic hopefuls.

In 2016 I favored Jim Webb in the Dem primary. If you know who he is you'll better understand my politics. And if you remember how he did (not very well) you'll appreciate my disenchantment with the state of the Democratic party.

Just because I don't experience the visceral hate and anger and fear the left has regarding Trump; please don't make the mistake that I'm not painfully aware of all his many faults.

fishfry Did you write this before the report came out that Trump ordered strikes (before cancelling)? — Michael

As you can probably verify from the timestamps, I did. In fact I speculated that he'd probably "bomb a couple of oil refineries," which is probably what he was planning to do till Tucker Carlson talked him out of it. Even Trump haters have to acknowledge that this week he's the sanest person in Washington. Among the Dem 2020 candidates, only Tulsi Gabbard advocates for peace, and she's polling at around 0.3%. What's wrong with the Democrats these days?

Today Trump said that he believes some rogue Iranian general made a mistake or did something stupid in shooting down the drone. It seems to me that this is conciliatory rhetoric, designed to justify NOT starting a major war.

Trump is the only one in Washington holding the line on peace today. Senator Lindsey Graham wants us to sink Iran's navy and bomb their oil fields. Trump's not doing that. He's saying he believes the drone shooting was not a provocative act requiring a deadly response.

Perhaps he'll blow up an oil field or two to placate the hawks. But what I see today is Trump remembering that he ran on a platform of opposing the stupid Middle East wars. I would say today is one day, and this is one moment, when even the diehard Trump haters should take a moment to reflect and to hope that Trump actually does have the coolest head in Washington right now.

I will agree with those who say he was wrong to abrogate Obama's nuclear treaty with Iran. That was a provocative act. But it doesn't mean Trump wants to start WWIII. Lindsay Graham does. I have no doubt Pompeo and Bolton do. But Trump, I think, is on the side of peace today. And I hope all the Trump-haters out there realize this; or at the very least, are hoping for the best.

Would Hillary be slow on the trigger? Her three-decade track record on militarism is appalling. You know that. So Trump was the peace candidate in the miserable choice we were given in the 2016 election. Trump called out Jeb and implicitly Bush 43 on Iraq. That's not nothing. It was unheard of at that time for a Republican to speak forcefully against the Iraq war.

I'm for peace. If you're for peace too, this would be a good moment to be on Donald Trump's side tonight as he talks down the generals.

The mathematics of biology involves a lot of differential equations, which are equations that show exactly how something changes over time. Perhaps you'd find them of interest.

https://en.wikipedia.org/wiki/Differential_equation

A related idea is that we can use differential equations to describe two interrelated changing systems. For example, when there are more predators than prey, the predators eat all the prey; then there aren't enough prey and the predators starve, reducing their population ... which allows the prey to survive and reproduce more, so that there are more of them to eat, and then the predators grow in numbers again. This eternal cycle of predator and prey populations is modeled mathematically by a couple of differential equations.

https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations

The law of identity, by the way, is not a law of mathematics. It's more primitive than that, it's a law of logic. Mathematics inherits the law of identity from logic; math doesn't posit or explicitly assume it.

The law of identity operates at a much "lower level" than that of modeling changing systems like weather or biology. The individual components of our model at any instant don't change; and then we can introduce a time variable to account for change from one moment to the next.

In fact basic calculus is the model here.

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change ...

https://en.wikipedia.org/wiki/Calculus

Another point of interest is dynamical systems, which Wiki describes as

... a system in which a function describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.

https://en.wikipedia.org/wiki/Dynamical_system

I've always felt that "dynamic systems" is more euphonious as a matter of English usage; but dynamical systems it is.

Change takes non-zero time, and a non-zero rate of change, for all non-infinite situations. The faster the change, and the greater the time over which it takes place, the greater the change that transpires; the greater the difference between the object at the initial state and the object at a later state. — Ilya B Shambat

Ah. Perhaps you already understand everything I wrote and you're making a more subtle point, which I'll take a stab at interpreting. You are right. In mathematical modeling, we model time by the real numbers, and then we say that "at time t" the world is in such and so state, no more and no less, unchanging as if frozen in ice. And at some later moment it's in a different state, so Newton showed us how to calculate the limit of this process as the time interval gets small, to thereby assign something we call the "instantaneous rate of change."

If perhaps you are pointing out that this is somewhat of a bogus or artificial abstraction, I quite agree. After all nobody knows whether time itself is accurately modeled by the standard model of the mathematical real numbers. That's a philosophical assumption made by science. It bumps into quantum theory. There are good reasons to doubt the mental model of static states as a function of time, and the standard real numbers as the official model of time. That viewpoint has been pragmatically successful for a few hundred years, but as to its ultimate truth, that's unknown.

I've been meaning to reply to this. I can't possibly respond technically on all you've written. You've outlined a research program that I may return to on an ongoing basis, but can't comment on at the moment. I'm grateful for all your references and comments. I know a lot more about HOTT today than I did before your posts.

It's interesting that you used Coq in school. You are a HOTT native. I of course learned my set-theoretic foundations long ago. As Max Planck said, science progresses one funeral at a time.

I don't know what is an isometry group — Mephist

The isometry group of 3-space is just the collection of all rigid motions of space. These are all the combinations rotations, translations, and reflections. Simple idea. They form a group in the sense of group theory, hence the name.

I only mentioned it because Banach-Tarski came up. BT is everyone's go-to example that proves that abstract math is completely divorced from common sense.

It turns out that the proof of Banach-Tarski is quite simple, at least in its outline. There are a number of steps but no step is particularly difficult. The core idea is encapsulated in the phrase, "The isometry group of 3-space contains a copy of the free group on two letters." When you unpack the meaning (which would take us far afield here) it is very simple and natural. No set-theoretic magic at all.

The Wiki article on the subject has a nice outline in case anyone's interested in demystifying this theorem for themselves. Point being that BT is much less esoteric than people think.

You cannot "build" an irrational number in set theory either. You can only assume that it exists because it's compatible with the rules that you use to build sets starting from other sets! — Mephist

I wanted to put in a word for the standard real numbers.

* Set theory's not needed for the standard reals. The idea of Cauchy completion arose historically long before Cantor's work on set theory. Cauchy completeness is a very natural idea, and the formality of defining the real numbers as the (equivalence classes of) Cauchy sequences of rationals would occur even to the most diehard constructivists. They might reject the real numbers from their ontology; but they could not deny its importance as an abstract idea.

The standard reals are the only model of the reals that are Cauchy complete.

The constructive reals are not complete. The sequence whose n-th element is the real number represented by the first n digits of the binary representation of Chaitin's Omega is a Cauchy sequence of computable numbers that fails to converge to a computable number.

On the other hand the other famous alternative model of the reals, the hyperreals of nonstandard analysis, are also not complete. Any non-Archimedean field (one that contains infinitesimals) is necessarily incomplete.

The constructive reals fail to be complete because there are too few of them. The hyperreals fail to be complete because there are too many of them. The standard reals are the Goldilocks model of the real numbers. Not too small and not too large. Just the right size to be complete.

* Cauchy completeness is a second order property. It's equivalent to the least upper bound property, which says that every nonempty set of reals that's bounded above has a least upper bound. It's second-order since it quantifies over subsets of reals and not just over the reals.

The second order theory of the reals is categorical. That means that every model of the reals that includes the least upper bound axiom is isomorphic to the standard reals. Up to isomorphism there is only one model of the standard reals; and it is the only model that is Cauchy complete. Even a committed constructivist would have to acknowledge these facts, and hold the standard real numbers as important at least as an abstraction.

* As a final point, I believe as far as I can tell that not every HOTT-er is a diehard constructivist. In some versions of HOTT there are axioms equivalent to Cauchy completeness and even the axiom of choice.

If I'm understanding correctly, one need not commit to constructivism in order to enjoy the benefits of HOTT and computer-assisted proof.

If John Bolton gets his way, yes. It's sad that Trump spoke so insightfully in 2016 against the endless, mindless semi-covert wars, and has now put neocon maniacs Bolton and Pompeo in charge of foreign policy.
— fishfry
What's so sad about?

Trump doesn't care a shit about anything else but himself.

For Trump to talk about endless wars was as hollow as his talk about fighting corruption, draining the swamp, was exactly just 'campaign talk', just something you say to get votes. Or do you really think someone wanting to build an even better military would really think about the endless wars? Heck, the guy was for attacking Libya. You had to be a idiot to believe this guy. So that makes a lot of people umm... well, you know. — ssu

If one doesn't regard an increased probability of a disastrous war with Iran as sad; one might be putting their politics ahead of their humanity.

Was I stupid to hope that perhaps Trump meant what he said about ending the ruinous wars? Certainly his militarism was a warning and a concern. But what was the alternative at the time? Hillary was a known warmonger. She was allegedly behind Janet Reno's disastrous attack on the Branch Davidians at Waco. She was behind Bill Clinton's war on Kosovo. Her vote for the Iraq war, along with her impassioned speech in favor of the war on the floor of the Senate, gave centrist liberals cover for supporting that war. Hillary could have stopped Bush, she chose to enable him. As Secretary of State she was behind the destruction of Libya, and got started on the destruction of Syria, which was completed by her successor John Kerry.

So if one was for peace, the choice was between Trump who at least talked the talk even if there were doubts he'd walk the walk; and Hillary, with a long track record supporting the neocon maniacs.

If you call hoping against hope for peace stupid, and you don't think the latest neocon attempts to start a major war in Iran are sad, I'd ask you to try to step outside your visceral feelings about Trump and try to figure out what you actually stand for. If it's peace, you better be sad at these latest developments.

If you'd rather see a war to prove you're right about Trump; than see peace and perhaps admit he was in the end less a warmonger than Hillary; you better check your partisanship. It's getting in the way of your humanity.

ps -- You may recall that Trump knocked out Jeb! by calling Jeb! out on Bush's invasion of Iraq. I didn't hear Hillary speaking out against any war, anytime, anywhere, in her entire career. If you want to make the case that Hillary is a great pacifist, I'll take the other side of that debate. Nor do I think Trump was lying. He has a crazy negotiating style. So far he's avoided getting us in any major new wars. So yes I'm hopeful in that regard. And note that it's the left's Russia hysteria that's endlessly ratcheting up tension in that direction. Just look at the NYT story yesterday about how the US is waging cyber warfare on Russia's power grid. Trump denies the story. Who's the warmonger here?

I see the same people who hated Trump's guts yesterday, finding yet one more thing to get hysterical about so they can hate him some more today.
— fishfry

I think the description of anyone critical about Trump as 'haters' is part of Trump's dishonest rhetorics. I mean, this man is threatening world peace, the global economy, the office of the Presidency, the environment, science, Republican party principles, the separation of powers, and so much more. He lies almost every time he speaks, he routinely flouts principles and demeans the office. So when this is pointed out, those making the comment are 'haters' - they 'hate Trump'. And 'hate' is something that demeans the hater. So it defuses any real criticism - 'oh, you're just a hater'. It's part of the way that Trump continually demeans the public discourse. Pity that people fall for it. — Wayfarer

This post so perfectly exemplifies the point I'm making that I have nothing at all to add. Except that claiming I characterized "anyone critical about Trump as 'haters'" is as disingenuous as can be.

I had a little insight. I read another Awodey paper, Structuralism, Invariance, and Univalence and it snapped everything into focus.

UA is the statement that "Isomorphic objects are identical."

In standard (set-theoretic) math, UA is a sort of folklore viewpoint. Everyone regards isomorphic objects as the same, even though we secretly know that they are different sets. We agree to ignore this little defect in set theory.

As Awodey puts it:

This common practice is even sometimes referred to light-heartedly as “abuse of notation,” and mathematicians have developed a sort of systematic sloppiness to help them implement this principle, which is quite useful in practice, despite being literally false. It is, namely, incompatible with conventional foundations of mathematics in set theory.

I internalized this idea a long time ago, and have been puzzled at what is the point of restating it in the context of HOTT.

The point is that HOTT allows us make the folklore viewpoint exact.

In HOTT we can express the idea that "Isomorphic objects are identical" as a precise mathematical statement. HOTT clarifies all the muddled conceptual areas that arise when we try to nail down how two different things are the same by virtue of sharing all of the "appropriate structural properties." What's a property? How do we know which ones matter to our structure? What's a structure? HOTT makes mathematically precise what was formerly vague. That's my latest understanding.

"HOTT makes UA legit" is the insight I've been looking for.

I found this article to be particularly useful in understanding the nuances of 'cultural appropriation', and I strongly recommend reading it. The author understands cultural appropriation as twofold: "first, an issue of cultural exploitation, and second, an issue of cultural disrespect". It does not mean that a culture "owns" something that cannot be adopted or re-purposed by another culture. — Maw

I'll read it, thanks. I do actually recognize what is meant by cultural appropriation in the context of disrespect and ignorance. I don't really think it's just eating burritos. I take your point. Still, what we read in the papers is the silliness of certain SJWs. Perhaps they'd make their own case stronger by not being parodies of themselves.

if my school had 'New York Pastrami Sandwiches' but it was served on potato bread instead of traditional rye bread, I would seek to have it corrected. — Maw

LOL, Well I've had plenty of "New York pizza" on the left coast, but never any actual New York pizza. But you don't actually go into Togo's or Subway and thoughtfully explain to them that the meat product they sell as pastrami is to actual pastrami as cardboard is to steak. Do you?

Putin is signalling Trump that's what he wants. — tim wood

LOL You guys will never give up. That Putin nonsense was invented on the night of the election by Robby Mook and John Podesta to deflect attention from how they managed to lose the most winnable election in history. How did the left sign on to this farcical red-baiting from the 1950's? I'm old enough to remember when the left instinctively distrusted the duplicitous bullshit coming out of the intelligence agencies.

Humor is, broadly, an attempt to demonstrate power over the subject of the joke. — yupamiralda

This is why we can't have funny comedians anymore.

So, basically, there is a criminal regime in the White House now. — Wayfarer

Didn't Hillary pay for oppo research written by a former member of British intelligence who got his (fictitious) information from Russian sources?

I watched the Trump clip. He said that if a rep of a foreign government told him something, he'd listen; and if he thought there was a problem, he'd talk to the FBI. That sounds perfectly reasonable. What is the president of the US supposed to do when a foreigner talks to him in private? That's the president's job, to listen to what a lot of people have to say about a lot of things.

Now, do I wish Trump had for once in his life transcended his worst tendencies and just put a sock in in? Yes. I wish Trump hadn't thrown all this red meat to his haters. Trump loves to throw gasoline on fires. I wish he'd dial it back.

And frankly, why did Trump let George Stephanopoulos, a loyal Clintonista, ask him anything at all? Hard to know. Also the optics of Stephanopoulos standing over him. The prez should have advisors looking out for things like that.

But I see the same people who hated Trump's guts yesterday, finding yet one more thing to get hysterical about so they can hate him some more today. Nobody's opinion of Trump was changed by this news item. Meanwhile someone's taking pot shots at oil tankers in the Strait of Hormuz. I'm a lot more concerned about that than I am about the latest leftist hysteria about whatever impolitic remark Trump made.

Oh dear, I hope you find a more appropriate place to put them — Baden

You seem emotionally invested in the topic. What most of us hear about the subject is college kids who demand that sushi be banned from campus menus, or that parties featuring Mexican sombreros are racist. What we hear about cultural appropriation mostly seems like childish acting out by spoiled leftists. If you can articulate some specific examples where it's really something that is evil, and perhaps give us some guidelines as to what's evil and what's silly, it would be helpful.

Some young woman wore a Chinese-influenced dress to the prom and the SJWs went nuts.

https://www.theguardian.com/commentisfree/2018/may/04/american-woman-qipao-china-cultural-appropriation-minorities-usa-dress

Can you give me a razor by which I can distinguish the serious from the silly? That would be helpful. Telling me to stick a burrito up my ass isn't helpful. Perhaps you can see that.

If John Bolton gets his way, yes. It's sad that Trump spoke so insightfully in 2016 against the endless, mindless semi-covert wars, and has now put neocon maniacs Bolton and Pompeo in charge of foreign policy. We'd lose a war with Iran, you know. They're not pushovers like Iraq, Libya, and the other countries we've invaded and destroyed

ps -- Watching PBS news. "Mike Pompeo offered no evidence but blamed Iran" for the attacks on the oil ships. That's how it works. If Trump doesn't fire these guys soon we'll be a full-fledged shooting war.

let's just bash a strawman, and by extension all things PC, because that's much easier and more fun than actually exploring the real damage to people and their way of life that can potentially be done by stupidly fucking with stuff that is very serious to them. — Baden

You're right. I'll stop eating burritos now.

Because if we assume we belong to a mind-independent world, then that world doesn't depend on our minds — leo

I think there's a step missing. The world is out there and I can use my body to affect it. But how did my mind affect my body? Searle makes this point in a video. "I tell myself I'm going to raise my right arm and my right arm goes up. How does that happen?" I'm paraphrasing his quote. But that's the real point. The question isn't how my mind can affect the physical world. The question is first, how does my mind affect my body? Once my mind has control over my body, I already have a physical instrument, namely my body, with which to affect the world.

Although I suppose if we say our bodies are part of what's "out there" in the world, then there's no difference. But my experience of my body is very different in nature from my experience of anything else in the world. For example I can not experience the pain of anything else other than my own body. I can have compassion and sadness and so forth, but I can never have an experience of the pain of anyone or anything else. So my body is different than the world; and therefore my point's valid. That the real mystery is how my mind can affect my own body, let alone anything else in the world.

Anyway, the idea of representing groups with functors is very simple — Mephist

Ah, group representations on steroids. I understand at least by analogy. The ncatlab link you provided was extremely helpful, thanks. Also the Wiki article on group objects is very good in case anyone else is trying to claw their way up this abstract area of math.

↪Terrapin Station OK, thanks for the suggestion. I'll read it — Mephist

LOL Now I don't feel so bad about my own ignorance of weighty matters!

Re the functor that cranks out groups ... I recalled and just looked up the fact that in some categories the forgetful functor Grp -> Set has a left adjoint which is the free functor; namely the functor that inputs a set and outputs a free object on it, such as for example the construction of a free group on a set. Could that be what you were thinking of earlier?

Can you consider this arguement against an infinite set,
What is the probability of an event happening over an infinite amount of events, it would be zero.We can go on and prove that the possibility of any event happening will be zero but that would be absurd if we applied it to the world. — Wittgenstein

Are you arguing against infinity in math? Or just in the world? It's perfectly clear that we all have a intuition of the natural numbers 0, 1, 2, 3, 4, 5, ... They are generated by the simple rules that:

* 0 is a number;
* If n is a number, so is n + 1.

Put those rules into a Turing machine and they crank out the endless sequence of natural numbers.

Nobody claims that the infinite collection [not yet a set, that requires the axiom of infinity] of natural numbers is instantiated in the natural world. It only exists as a mental abstraction, like justice or traffic laws or Captain Ahab.

Or are you arguing that you accept mathematical abstractions but denying that they're physically real? That's perfectly sensible.

But by referring to the world, you are out of the realm of the abstract and instead making a trivial point about the world. That weakens your argument considerably.

In any event, infinitary probability theory is well understood and allows for probability zero events that nevertheless may happen. For example the probability of picking a random real number and having it be rational is zero; yet the rationals are plentiful. [That's not a precise statement, but it can be made precise without loss of intuition].

The category of modules (https://en.wikipedia.org/wiki/Category_of_modules)

and the category of vector spaces (https://ncatlab.org/nlab/show/Vect) for complex numbers — Mephist

I understood pretty much everything you wrote. I know what a group object is (as of this morning) by looking it up in Awodey's most excellent book on category theory, which I've been lazily working through (ie "glancing at") for a while now.

So your functor remark was only referring to the forgetful functor that maps a module or vector space to its additive group? Am I understanding that correctly?

Why is the word “cuff” just fine, but pronounced backwards, considered offensive? — Frank Apisa

As George Carlin said, you can prick your finger, but you can't finger your prick.

The group of integers mod 4 and the powers of i are not objects in the category of groups, but functors from categories with the appropriate structure of morphisms to the category of groups. You can still use functors to build whatever structure you want, and in the category of functors, these structures are no more isomorphic. — Mephist

Ok I can't comment on type theory at all because I'm totally ignorant in that area.

But I'm a little confused by this paragraph. I would have thought -- in fact I do think -- that each of these groups is an object in the category of groups; and that there are morphisms from one to the other and back whose compositions are the respective identities, hence the two groups are isomorphic. This is true in category theory.

I would think that UA simply says that from now on I should just think of there being one group, the cyclic group of order 4, and I shouldn't care that there are two different set-theoretic presentations of it. Which is how we normally think of it anyway. Which is how I'm approaching UA.

But your para is quite opaque to me. You say the integers mod 4 and the powers of i are NOT objects in the category of groups. I'm in deep trouble here, please throw me a lifeline.

But to see them as functors, I'm not sure I get that. Functors from "categories with the appropriate structure of morphisms to the category of groups." Ayyy. I'm in trouble. Help me out please.

What are the categories with the appropriate structure etc.? Can you give me an example?

And what can you mean in the first place by denying that these are objects in the category of groups? Maybe I know a lot less than I think I do. Which is quite possible. But I know what a categorical isomorphism is, and it requires that there are objects that are isomorphic. So I am pretty lost in your paragraph.

God doesn't have Moosehead Ale, eh?

Transfinite ordinals divide into those which are specified constructively as tree-growing algorithms and those which denote unspecified trees to be supplied by the environment, whether bounded or unbounded. — sime

You have a link in support of your ideas? They are very strange. I don't want to flat out say they're wrong, since my ignorance is vast. But I know a little about ordinals and I can't correspond your words to anything I know.

infinity is equivalent to volatile and unbounded. — sime

I don't know what is a volatile and unbounded set. Can you provide some examples so I can understand what you're saying?

Volatile is not a term of art in math at all. And its use in C programming is very specific as I think we agree. It just tells the compiler not to optimize the variable.

The integers are unbounded because you can't draw a finite circle around them all. The unit interval is bounded since all its elements are within 1 unit of each other. Yet the unit interval has far larger cardinality than the integers. So I am not sure what you're trying to get at.

Regarding ordinals, you say that there are some which "denote unspecified trees to be supplied by the environment ..." That's .... well again it's idiosyncratic. There are ordinals which are computable and ordinals which are not. Are you thinking of the Church-Kleene ordinal or one of the other exotic countable ordinals?

I really don't know what you mean by saying (some) ordinals are supplied by the environment. That's not what ordinals are. Ordinals are order types of well-ordered sets. They don't take on values like memory locations in an executing computer program. I think perhaps you might be trying to push programming analogies farther than they can go.

In C programming, the equivalent symbol to infinity is the volatile keyword. — sime

Jeez that's not true. A volatile variable is one that is, for example, mapped to an external data source. Declaring a variable volatile tells the compiler that it can't depend on nearby code statements in order to optimize the variable.

This simply has nothing at all to do with transfinite ordinals and cardinals as understood in math. It's apples and spark plugs.

it also fails to discriminate sets which are volatile and bounded from sets which are volatile and unbounded. — sime

Those terms have no referents in math. I am not sure where you are getting these notions. Your ideas about the axiom of choice and mathematical infinity are idiosyncratic to say the least.

P.S. Isomorphic structures to be equal in Set would mean that you don't distinguish between different implementations of the same algebraic data structure. This is good for mathematics but not for programming. — Mephist

Is the UA intended to apply to standard set theory? In other words in the category of sets, a morphism is a bijection. If isomorphism is identity, then there is exactly one set of each cardinality. This is actually a very helpful point of view sometimes. For example the set of natural numbers and the set of rational numbers are exactly the same set. All we've done is named its elements in two different ways. That's my set-theoretic understanding of "isomorphism is identity."

Another common example is, say, the group of integers mod 4. This group is isomorphic to the set {1, i, -1, i} of complex numbers, with the operation of multiplication. These are two different groups, but they are isomorphic to each other. This is something students of abstract algebra come to understand at some point. That we can mentally identify the two groups as if they were the same group.

On the other hand there might be applications in which we do want to remember that these are two distinct groups that happen to be isomorphic. From the traditional (non-UA) point of view, we say that there is only one cyclic group of order 4; but when pressed, we admit that it shows up in many different disguises that are set-theoretically distinct.

Is this how I should be approaching trying to understand the UA?

There is an infinite possibility of line segments with different lengths but if we were to join them, we would never complete the task — Wittgenstein

Having already admitted the existence of uncountably many line segments, you can take their union in one application of the axiom of union to join them. It's hard for me to understand admitting the existence of a collection of sets but denying their union. Of all the axioms of set theory that are commonly debated or questioned (choice, powerset, infinity, foundation), union is not one I've ever heard anyone question.