it's pretty simple... we have many billions of people and the number is growing exponentially at an accelerated and unsustainable rate. — Lif3r

A neo-Malthusian.

Malthus was wrong. Human ingenuity wins. We crawled out of caves and built all this. Don't sell humanity short.

Remind me of the defs. A section is a right inverse as I understand it. There can be a lot of right inverses to a function, you just keep choosing different elements in the preimages of points. Is that the bundle?
— fishfry

Yes. — Mephist

Ok a fibre bundle is the collection of all possible right inverses to a function. And a section is one of those right inverses. Yes?

I may be off the mark here but I wonder if it's relevant that we have introduced a little bit of nonconstructivism. How do you know you can always take a section of a surjective function? That claim is equivalent to the axiom of choice. It makes sense. Take the cosine or sine functions. Each real number between 0 and 1 has infinitely many inverses, separated by 2 pi. For each point in [0,1] we choose an element from its inverse image. So you can see how the axiom of choice comes up. Of course in this particular case we could take the smallest positive inverse, so we don't need the axiom of choice. But in the general case we do.

Where does measure theory (surely not taught in high school) intersect any of this? I've used it in various integration processes, the most interesting being functional integration. And Feynman constructed his sum of paths integral in more or less that concept. — jgill

Oh, it doesn't. I mentioned that the computable numbers have measure zero in the space of bitstrings, and @Mephist asked me how that's defined mathematically, so explained it a little. Then he responded by saying he's known about this since high school. So @Mephist I apologize if I misunderstood you.

The subject came up in the context of my perennial hobbyhorse that there aren't enough computable numbers to make up a decent continuum; and why aren't the constructivists ashamed of themselves. I have never gotten an answer to this question that satisfies me. At some point they say "Martin-Löf type theory" and I know I've lost the argument. It's a cult. [mild humor intended].

No, I like "normal" mathematics: no computers involved. But having a theorem-prover as Coq to be able to verify if you can really write a proof of what you think is provable is very helpful. — Mephist

I'm reading the data science paper you linked. They teach sheaf cohomology to data scientists. That is so fascinating. To me, with my math background, sheaf cohomology is something that would take someone a long time to learn, at least a couple of years of grad school or more. But at the application level, the concepts have filtered down and you don't have to actually know any of the original mathematical context in which these ideas evolved. It's yet another illustration of the applicability of highly abstract math. The "unreasonable effectiveness" all over again. Category theory dates from the 1940's but it's only peeking its head into the real world in the past twenty years.

Yes. — Mephist

I have no idea what question is was a response to. But if you agree with me, you have excellent taste!

Yes! I should have added a formal definition, but I have an aversion to writing symbols on this site :confused: I added a link with a clear picture, I think. — Mephist

I think when I grok how fiber bundles can be likened to proofs, I'll be enlightened.

Did you learn all this from the CS viewpoint? Just wondering.

So, there is a main ingredient that is missing: points! Topos theory is a formulation of set theory where sets are not "built" starting from points. Sets (the objects of the category) and functions (the morphisms of the category) are considered as "primitive" concepts. The points are a "secondary" construction. — Mephist

Ok now I know this idea as ETCS: The extended theory of the category of sets, which is an implementation of set theory on top of category theory. Is this the same thing as what you're talking about?

Yes it's wondrous that we can do set theory without talking about points! Deep philosophical implications since we no longer care at all what a thing is, only how it relates to everything else.

OK, I'll stop here for the moment, because I am a little afraid of the answer "that's all bullshit, I'll not read the rest of it..." — Mephist

No this is totally fascinating, very clear writeup, worthy of study. I know what fiber bundles and sections are. I can't quite grok the application to proofs but it may come to me.

Remind me of the defs. A section is a right inverse as I understand it. There can be a lot of right inverses to a function, you just keep choosing different elements in the preimages of points. Is that the bundle?

OK, never mind. Sorry for continuing to repeat the same things! — Mephist

No problem, you've gotten me interested in sheaf theory and then on to topos theory. But I'm still probably more oriented to the mathematical applications than the logical ones.

A sheaf is a topos at the same way as a set is a topos: it's the "trick" of the Yoneda embedding! :smile: do you understand now? (sorry: bad example.. let's say that a sheaf can make everything - more or less - "become" a topos) — Mephist

I'm going to spend the next week working through this material. I'm encouraged that I can understand what a sheaf is and know a few examples; and I know enough category theory to get by.

May I ask you a question? How does one come to know this material and not have heard of measure theory?

A sheaf S over a topological space X is a "fiber bundle", where the fibers over a point x in X are disjoint subspaces of S. Now, a section of the fiber bundle (https://en.wikipedia.org/wiki/Section_(fiber_bundle)) is what in type theory is called a "dependently typed function", that from the point of view of logic is interpreted as the proof of a proposition with a free variable x: the fiber bundle is the proposition (that depends on x) and a section of that fiber bundle is a proof of that proposition. — Mephist

Now that is interesting, because fiber bundles are a big thing in differential geometry. It's interesting that they lead directly to type theory. Thanks for pointing that out.

As with your other technical post, I'll defer comment for now but I'll read through them.

A section is a proof ... a section is a proof. I can't quite see that. I know that a section is a right inverse of a function. For example if $f(x) = x^2$ then a section is a function that, for each nonnegative real, picks out one of its plus or minus square roots. Is that about right? The section is the right inverse, so it's essentially a choice function on the collection of inverse images of all the points. Do I have that right? How does that become a proof?

it's the formulation of set theory in terms of objects, arrows, and universal-mapping properties. — Mephist

That's what EVERYTHING is in category theory! So that didn't tell me anything about topoi!

You wrote me two detailed technical posts that I'll try to catch up to later. But I'm actually on my own path. I found a nice paper on sheaf theory that I can understand and I'm working through that.

I've also in the past seen a video with Steve Awodey where he worked out the definition of a Cartesian closed category then applied it to logic. So I don't even think you have to go all the way to topoi to get to applications in logic. I'm probably going to spend a few days reading papers and see if I can get a hold on a vertical slice of understanding.

Yeah, I see. We come back to the same issue. Tarski was a great logician but also a great algebraic geometrist. Some people have already tried to explain to me why it is apparently one and the same thing, but that hasn't registered with me already. I still fail to see the "obvious" link between both. — alcontali

Yes thanks. I was trying to make this point to @Mephist the other day and this is a good example. The connections among topology, algebraic geometry, and logic have been studied since the 1930's. This stuff's been around for a long time.

Wish I could say more about L-S but I don't know much about it.

But I can give you the reference to a very good book (in my opinion) on this subject that is easy to understand for somebody that has some basis of category theory:
- Title: "TOPOI THE CATEGORIAL ANALYSIS OF LOGIC"
- Autor: Robert Goldblatt — Mephist

Thanks. I know Goldblatt as the author of Lectures on the Hyperreals. I'll add this book to my growing list of books to read someday. So far the list stretches into several of the next few lifetimes.

I am reading articles on sheaf theory (of the mathematical kind) and it's very straightforward, although most of the serious applications are beyond me. I will read the paper you linked about network topology, that example was very insightful. Steve Awody's book on category theory has a lot of applications to logic. Another book on my list.

I think I'm missing more than a "few" molecules but that's beside the point. What I want to know is whether the distance AB is the same as the distance BA where A and B are the same points. — TheMadFool

If they are points in an abstract mathematical metric space, yes. If they are physical points, no, for two reasons. One, the earth is constantly changing shape. The effect is tiny but you're asking if the two distances are exactly the same. Second, you have two measurements. Each is only an approximation. You could and most likely would measure two different lengths, that are within error tolerance of each other.

Of course for all practical purposes, we regard the two distances as the same. Why are you asking?

If the past stretches to negative infinity from the present wouldn't that mean the universe would've to experience positive infinity to reach the present? — TheMadFool

No. The past does not stretch to negative infinity any more than 1/x is defined at negative infinity as in your other thread. It's the same model of the integers. Or if time is continuous, the real number line. It doesn't start anywhere. It just is.

If B = past and A = the present then the time AB = negative infinity and the time BA = positive infinity. — TheMadFool

No. Just as in your 1/x thread, there is no point at -infinity. There is no left hand endpoint to the number line, nor a right hand endpoint. And you see on the number line a point marked 2020? That's where we are. How did we get here? Nobody knows. It's a great mystery. But I see no reason that there must, by logical necessity, be a leftmost point on the real number line. Mathematically there isn't. Nor do I see why time or causation should be any different.

If you agree with me so far — TheMadFool

I disagree with your thought process entirely. There is no point at minus infinity, either in the integers, as in this thread, or in the real numbers, as in the 1/x thread. You have an incorrect picture of the integer and real number lines. They keep going forever to the left and to the right. It's perfectly symmetric.

If the universe is eternal, then your model fails. Would you agree with that?

So you are making a metaphysical claim, that the universe (or time, or causality, etc.) is not eternal. That's an opinion. You can't possibly claim to know for sure unless God sent you an email about it.

and I see no reason to not do so then that would mean a positive infinity of time should've elapsed to reach the present i.e. a completed infinity is require and we know that completed infinity is an oxymoron or, to be explicit, a blatant contradiction. However, I keep an open mind about this: there are more things in heaven and on earth than can be dreamed up in your philosophy — TheMadFool

Your premises are wrong, your reasoning is wrong, your conclusion is wrong.

But I hope you will answer my question: How do YOU know that the universe is not eternal? Has God spoken to you? Does she have a hot tip for the Super Bowl?

It is not only from celebrity physicists that philosophy gets a bashing. Philosophers themselves also appear very critical of philosophy, which seems to be self-contradictory, but is it really? — Pussycat

Great point. Everyone likes to make fun of philosophers!

We are talking about the origin of everything; IE huge amounts of matter; IE a macro, not micro problem. In the macro world the cause always comes before and determines the effect. — Devans99

Did you get that from God's lips to your ear? You have an opinion, nothing more.

A. Assume an infinite causal regress exists
B. Then it has no first element
C. If it has no nth element, it has no nth+1 element
D. So it cannot exist — Devans99

C is confused. The integers have no first element. But every element has a successor. For every n there's an n+1. It does not have "n-th elements" because it's not a well-ordered set. There's no fifth member of the integers. What of it?

I'm surprised at this attitude, although it continues to surprise me how little concern there is for climate change in the members of the forum from the US. Is it a partisan stance perhaps, I recollect Trump's insistence that climate change is a Chinese plot, a deception to persuade the west to ruin its economies and competitiveness. — Punshhh

i noted that many environmentalists are for population control of third worlders. Nobody ever asks the third worlders what they think. Some extreme environmentalists are anti-human; and I oppose that type of environmentalism.

I can be for clean air and water without wanting to deprive the third world of their aspirations to a better life.

If we represent xy=1 as a predicate function γ(x,y)γ(x,y) which is true when xy=1 and false otherwise, then we get a model-theoretical model with logical sentences that are true or false about (x,y) tuples. — alcontali

Surely this is true about the zero-set of any function whatsoever. The study of the zero sets of polynomials is algebraic geometry. That's where I'd look for answers to these sorts of questions.

Off the top of my head since there must be models of the reals of all infinite cardinalities, the zero-set of xy - 1 = 0 would have the cardinality of whatever model you're looking at. But honestly to the best of my understanding I don't think this means anything. My personal opinion is that people shouldn't get too hung up on Lowenheim-Skolem. It's essentially a curiosity. On the other hand, Skolem thought that it showed that the concept of set isn't very well defined. He may have a point.

OK, I'll try to explain this point.
The fact that there is a relation between topology and logic (mediated by category theory) was well known even before, you are right. But Voevodsky's "homotopy type theory" (https://homotopytypetheory.org/) does not say simply that there is a relation between topology and logic: it says that "homotopy theory" (that is a branch of topology) ( https://en.wikipedia.org/wiki/Category:Homotopy_theory ) and Martin-Lof intuitionistic type theory with the addition of a particular axiom (the univalence axiom - https://ncatlab.org/nlab/show/univalence+axiom) ARE EXACTLY THE SAME THING (the same theory). Meaning: there is this axiomatic theory that speaks about homotopy between topological spaces, expressed in the language of category theory (and then in ZFC set theory - it is still valid in any topos, but I don't want to make it too complicated). So, the terms of the language are spaces, points, paths connecting points, equivalence classes between these paths etc...
Now, if you take whatever theorem from homotopy theory and RENAME all the terms of this theory, substituting the word "types" to the word "spaces", "proofs" to the word "points", "equalities" to the word "paths", etc... (lots of details omitted, of course), you obtain a theorem in type theory. And if you take any theorem in type theory you can reinterpret it as a theorem about topology. — Mephist

Be advised that any paragraph containing the name Martin-Löf instantly glazes my eyes. I've had all these conversations too many times. I totally believe everything you say but can't actually figure out what point you are trying to make. I don't disagree with anything you say, and I'm aware in varying degrees with various aspect of the things you talk about. I just don't know why you're telling me this. I don't disagree with you on any of it and I can't relate this to whatever we are talking about.

Can you just remind me what is the point under discussion?

Thanks for saving me the effort of looking it up. That one sentence is enough for me. — jgill

I do not think this is so bad. I'm learning what a sheaf is and after that, topoi are the next step up. If I figure anything out I'll post it. What's interesting is that the highly super-abstract algebra has deep repercussions in logic. Even without any of the details, that's the takeaway.

Infinity is not a number and even if it is 1/(-/+infinity) will always be a non-zero value for the simple reason that there's no number that satisfies the equation 1/x = 0. Dividing by larger and larger x values will result in 1/x approaching zero as a limit but it'll never be the case that 1/x = 0. — TheMadFool

There are no points at infinity on the real line, so the function's not defined there. And just because a function has a limit at infinity, that does NOT imply that the function is defined "at infinity," which is meaningless in the real numbers.

Is that what you are saying?

It is true that they can be very abstract objects in mathematics, but for a data-science person a sheave is mostly a data-correlation tool. A sheave can represent a cellular-phone network and relate each cell of the covered area with the set of users that are connected to that cell. — Mephist

OMG that sort of makes sense. Thank you for that example. I've been reading up on sheaf theory and every presentation that comes up on Google is heavily mathematical, so much so that I can't for the life of me understand how they're teaching this stuff to undergrads in non-mathematical fields. I will definitely read the link you supplied. Perfectly sensible ... to each individual cellphone you associate their call network. So it's an algebraic structure -- a network or graph -- assigned to each element of some set. Very nice example. And of course a great illustration of how wild mathematical abstractions so often end up being incredibly useful in the real world.

And a topos is an abstract sheaf? And what I read was that topoi are inherently intuitionistic. I haven't followed that chain of reasoning yet.

If I travel from Istanbul to New York by plane the distance is 8,065 km. If I return from New York to Istanbul, again by plane and on the same route the distance will again be 8,065 km right? — TheMadFool

Uh ... yeah, is this a trick question? I don't see the relevance. But yes, I'd say so. Of course tiny fluctuations in the shape of the earth mean that the distance wouldn't be exactly the same. And since all measurement is approximate, we can never know for certain if the distances are the same!

It's a rule of the mathematical idea of a metric space that the distance from A to B is exactly the same as the distance from B to A. But in the real world it's an approximation. You might be a few molecules off.

I was simply pointing out that, taken as a function, f(x) = 1/x, we can see that just because f(a) = f(b), it doesn't imply that a = b. — TheMadFool

On the contrary. We can exactly conclude that if f(a) = f(b) then a = b in this case. There are no endpoints or values at infinity. A function need not be defined at a point in order to have a limit there. Just because the limits are equal doesn't violate injectivity because the function's not defined at infinity. In fact the points at infinity don't exist.

As a relationship, and you told me about it in another thread, it's a case of injection where both f(+infinity) and f(-infinity) give the same result 0. — TheMadFool

A function is an injection when that CAN'T happen. An injection is exactly when $f(a) = f(b)$ implies $a = b$. And this is exactly the case with $f(x) = \frac{1}{x}$.

You can see this visually by applying the horizontal line test to the graph of the function. Every horizontal line intersects the graph in at most one point. The x-axis doesn't intersect the graph at all.

Compare this to the graph of $f(x) = x^2$, a parabola pointing upward. Every horizontal line except the x-axis crosses the graph at two points. So it's not injective.

I take this to mean that the end behavior of f(x) = 1/x is very much like g(x) = x^2 in which (-a)^2 = (+a)^2 but -a not= +a. — TheMadFool

No, exactly not. The points at infinity aren't there. The function has the limit zero at plus and minus infinity, but the function is not defined at those points, nor do those points exist.

What is true is that $\displaystyle \lim_{x \to \infty} \frac{1}{x} = 0$ and $\displaystyle \lim_{x \to - \infty} \frac{1}{x} = 0$. We could colloquially or informally say "$f(\infty) = 0$" but only when we realize we are using a shorthand notation for the respective limits.

But in your case you are taking this notation literally, and that's incorrect. The real line does not include any points at infinity or endpoints. So $\frac{1}{x}$ is in fact injective; and your notation is wrong. There are no endpoints to the real line and the function is only defined on the real numbers.

Yet, simple algebra does show that if 1/x = 1/y then x = y. — TheMadFool

Yes! You just proved that $\frac{1}{x}$ is injective. You have that totally right.

You're just confusing yourself by imagining the function is defined at the "endpoints" of the real line. But there are no endpoints of the real line. They don't exist. The notation $f(\infty) = 0$ is only to be taken as a shorthand for the statement $\displaystyle \lim_{x \to \infty} \frac{1}{x} = 0$, which has a very specific meaning.

In fact there is no real number $x$ such that $\frac{1}{x} = 0$. That's perfectly obvious on its face.

The function f(x) = 1/x doesn't involve squaring but we do multiply by the product xy which is (-infinity)(+infinity). Is this where the problem occurs? — TheMadFool

No, there is no problem. There are no endpoints and the function's not defined at these nonexistent endpoints. What is true is that two limits happen to both be the same; but that is NOT a failure of injectivity, because the function never actually hits zero.

If there's a tl;dr here it's this: A function need not be defined at a point in order to have a limit there. And it need not be defined "at infinity" in order to have a limit at infinity. $f(x) = \frac{1}{x}$ is injective.

The correspondence between topology and logic instead, that's one of the most popular and ideas of today's mathematics! — Mephist

You mentioned topos theory in one of your posts. I read the Wiki page, or re-read it since I've looked at it before. It's abstract sheaf theory. What's sheaf theory? It's the idea of assigning an algebraic object to each point of a topological space or manifold. For example the set of continuous functions that vanish at a given real number is an ideal in the ring of continuous functions on the reals. So the ideals of the ring give you information about the points. This is a fairly sophisticated mathematical point of view.

And then topos theory is abstract sheaf theory. So mathematically, this is advanced grad student level. But apparently a lot of the terminology and concepts are trickling into computer science and other fields.

I was wondering in what context you'd seen topoi. I know there's a lot of category theory in computer science these days. But my sense is that topoi are fairly sophisticated mathematical objects, at least in their mathematical applications.

The wager isn't a logic flaw. If one could form a belief by flipping a switch, it would make sense for anyone who thinks there's at least a small chance of a god who rewards us after death for believing in him. Switching to believer costs you nothing, and it at least has that small chance of benefitting you. So the problem is that beliefs don't work that way. — Relativist

What I'm saying is that the wager depends on a god who hands out eternal reward for believing, and eternal damnation otherwise. That's a vindictive god, a petty god. A god unworthy of the name. If you decided you didn't believe in me, would I smite you? Of course not. I wouldn't care one way or another. But if the Christian God finds out you don't believe in him? Eternal damnation. How childish!

In order for Pascal's wager to make sense you have to believe in such a god. And that's not a plausible idea. The creator of the universe is not that petty. The creator of the universe accepts my belief or disbelief in him with equanimity either way.

The God I believe in is not a God that gives a shit whether I believe in him. Thus I refute Pascal's wager.

and more importantly how and when to not populate) responsibly — Lif3r

The first world environmentalists (are there any other kind?) always want the poor to stop making babies and not demand a modern standard of living.

Would you wreck our economy for a nebulous tenth of a percent temperature change that might or might not be just a statistical artifact of your model and not reality?

Celebrity physicists bashing philosophy is as old as Feynman if not older.

and even Martin-Löf type theory has a lot of variants (too many to be something important, right? :smile: ) — Mephist

(Eyes glaze).

But I think that now the picture is becoming quite clear (even thanks to Voevodsky's work): — Mephist

Name drops Voevodsky. Didn't we do this last week? Or was that someone else?

there is a very strict correspondence between topology and logic. — Mephist

Perfectly well known before Vovoedsky.

But you have to "extend" the notion of topology to the one of topoi (a category with some additional properties). ZFC is the logic corresponding to the standard topology (where lines are made of uncountable sets of points). But ZFC and the "standard" topology are not at all the only logically sound possibility! (that in a VERY short summary) — Mephist

Do you know topos theory? Can you "explain like I'm five" to someone who knows a little category theory?

But when did I ever say standard math is the only logically sound way of doing things? I don't think I hold the view you're trying to dispel.

Yes but mathematics needs computations for proofs, and computations are physical processes. — Mephist

I was reading through the thread and this caught my eye. Why does mathematics need computations? Because before there were computers, the humans did the computations. In math we still do. Computations with pencil and paper are no different in principle than the same computations done on a supercomputer, and vice versa.

So in fact mathematics itself, defined as everything mathematicians have written down or even thought about -- since thought is a physical process too -- is a "physical process." The work of mathematicians takes energy; and therefore it should be possible to study the nature of math as a physical process.

Or as the saying goes: Mathematicians are people who turn coffee into theorems! I think that pretty much captures the spirit of the idea.

In other words we have a single output for two inputs that are the very name of being poles apart. — TheMadFool

So what? The cosine function has infinitely many inputs that go to the same output. $\cos \theta = \cos(\theta + 2 \pi n)$ for any integer n. And they are spread out arbitrarily far apart. What of it?

Just because you have two quantities that happen to have the same limit, doesn't mean that the two quantities are equal to each other. Just like two different travelers who both end up in Poughkeepsie. They aren't the same person just because they ended up in the same town.

When one considers the function ... — TheMadFool

Did you find the picture helpful?

P.S. Here's a citation taken from wikipedia: — Mephist

So the lambda formulation is more granular, able to support more nuanced theories? Something like that?

Anyway I know about Coq and the proof assistants and such, but my eyes glaze over every time I read the phrase, Martin-Löf type theory. So I know all about the existence of all this stuff, without necessarily knowing much about it. On the other hand, on something like homotopy type theory, I don't know anything about types. But I do happen to know what homotopies are, and that gives me a little insight into what they're doing.

I don't know what would be the equivalent limitation to Turing machines that corresponds to dependently typed lambda calculus (if there is one). So, I should have said that we can assume that the "original" (non limited) Turing machine does not exist — Mephist

You're saying TMs don't exist but finite state machines do? Maybe so, but then you'll make your physics a lot harder if you can't even run an algorithm to approximate a constant of nature. I never studied lambda calculus so I can't comment on the rest. But since lambda calculus and TMs are equivalent, they either both exist or neither do. As far as I understand.

if we travel from our past which is negative infinity to the present, point 0 on the integer number line, then we would have to traverse a positive infinity of time to reach the present, point 0 on the integer number line. However, positive infinity is, by definition, an interminable quantity and a task that cannot be completed. — TheMadFool

If you allow infinite movement to the right, why not to the left? The situation is perfectly symmetrical except for your irrational attachments to false and confused beliefs about time.


To get to the number 5, we have to traverse infinity from the right! That's exactly as sensible as what you're saying. You are saying that left and right are asymmetrical. That's a belief, not a fact.

Is infinity a Western Concept? I wasn't aware of that? Anyway...here's a simple argument: — TheMadFool

As Bernie said to Liz last night, "I did not ever say that!"

I said that the concept of the world as having had a distinct moment of beginning is a western concept. It's not even my idea. Someone else replied to one of my posts noting that Buddhists would have a very different concept of time and a different mathematical model. So the idea that causation is a well-ordered collection, with a first element, is an assumption that derives from the West's Christianity. That seems to be a nice explanation for why people cling so deeply to the idea that "there can be no infinite regress." As a math major I immediately think of the integers. They have infinite regress.

And for that matter, why are the people opposed to finite regress not also opposed to infinite progress? Why shouldn't there be a maximum integer going to the right? After all to get to 5 from the right we have to get to 6, but first we have to get to 7 ... and that process could never start. So there must be a maximum integer.

You see how ridiculous that argument is. But if you go from left to right, suddenly it's meaningful?

Says who?

But more to the point ... how on earth did you misquote and misconstrue me like that?

You are failing completely to understand the dynamics of causal regresses. I have given you examples that I child could follow. I am almost at a loss. — Devans99

I just happened to run across this article this very morning.

In the quantum realm, cause doesn’t necessarily come before effect

https://www.newscientist.com/article/mg24532650-700-in-the-quantum-realm-cause-doesnt-necessarily-come-before-effect/

People's mental model of there being a first moment of time then a next then a next and always going in one direction, is something they picked up when they were eight years old. The very idea of sequential time doesn't even hold up to the scrutiny of modern physics.

There is simply no reason at all that time and causation couldn't be modeled as the integers. Or maybe even as the real numbers ... one moment smearing into the ones nearby, with the concept of "next" being nonexistent.

How do you know causality is discrete at all? Maybe it's continuous. You have no way of knowing what's true. You just cling to an outmoded idea because you won't step back a level of indirection to see the perfectly reasonable alternatives. Buddhists don't agree with your concept of time. Quantum physics doesn't agree with your concept of time.

Why won't you recognize that you have and opinion, and not a fact?

But what I am saying is that you can equally well assume as an axiom (that would be incompatible with ZFC) that Turing machines DO NOT exist! — Mephist

I've never heard of this idea, that TM's don't exist. I see no problem expressing TMs in set theory. An unbounded tape of cells is modeled as the integers. You have some rules that let you move right or left. It's pretty straightforward.

Can you say more about this? I have never heard this idea at all. It seems VERY restrictive. Perhaps it's like denying the axiom of infinity. Logically consistent but too restrictive to do math with.

I'm not even sure what that means, that TMs don't exist. The Euclidean algorithm to find the greatest common factor of two integers is 2400 years old. That's the world's first algorithm. It's a Turing machine. It exists. I think "TMs don't exist" introduces a contradiction.

Constructivist theories correspond to elegant constructions in topology, represented as internal languages of certain categories. In comparison, ZFC axioms seem to be much more arbitrary, from my point of view. — Mephist

I'll stipulate that a lot of very clever people are on the neo-intuitionist bandwagon these days and that their viewpoint represent the future, while mine represents the past; at least for now. My unease with constructivism is psychological. I love the catechism of standard classical math. It's a matter of faith, not science. Or if faith is too strong a word, familiarity and preference.

If a computation is too long to be performed by any computer even in principle, is it still valid? — Mephist

A TM with a program too long to write down in the age of the universe is still a TM. Practical resource limitations do not apply to the theory of computability.

No computation is too long to be performed "in principle." In principle a TM is a finite sequence of instructions. No matter how long it is, as long as it's finite it's computable in principle, if not necessarily in practice.