But then these numbers - growth constants like e and phi – are ratios and so are dimensionless unit 1 values more than they are some weird real number. — apokrisis

To correct myself, e and phi are different in that e is transcendental and phi is algebraic. That's something I need to dig into further.

My argument is that the whole potential vs actual infinity thing comes from the fact that our ideas about numbers are based on systems of constraints. We want to close some space of values in a way that creates a set of numerical objects with shared properties. And so we end up with a set of "all the integers" that is a class of objects with tightly specified properties. And then relax some of those constraints and we find the integers nested in the larger set that is "all the rationals".

This is a subsumption hierarchy in hierarchy theory. Each successive set becomes larger as you relax the constraints that form its boundary. So from the point of view of the integers, they are potentially infinite when seen from inside their closed world - a closed world that is unbounded in its openness as an integer generator – and then become viewable as actually infinite in being a crisply restricted domain within the larger and more relaxed space that is the set of rationals.

So virtual vs actual infinity is a point of view. Each level of number can contain within it a more constrained notion of number. And also itself be contained within a less constrained one.

Each such set is a machine for cranking out sequences according to some set of rules. It is a symbol generator, and so is open in that it possesses the finite means for spitting out infinite steps. But then from the point of view of some less restricted generating algorithm, the set looks to be closed in terms of all the values that can be produced in that way. There is now some larger set of values that becomes possible by a generator with less constraints producing values with less intrinsic structure.

This nested view is illustrated in diagrams like this...

Adding to my response about the particular paradoxes. Even if we granted that they indicate flaws in the concept of infinitude, then that is a concept of infinitude extended beyond set theory into imaginary states of affairs for which set theory should not take blame. Those paradoxes don't impugn set theory itself. — TonesInDeepFreeze

I would say that those paradoxes don't necessarily impugn set theory itself. Just as actual infinities (rather infinitesimals) were banished from (mainstream) calculus and replaced with potential infinities (through limits) leaving the applied math essentially uneffected, I wonder if something similar could be done to the actual infinities of Set Theory.

I know so little of cosmology that I don't know how to dispel my bafflement that the universe could be finite or my bafflement that the universe could be infinite. — TonesInDeepFreeze

I think it's quite easy to imagine a closed finite universe, for example a sphere of finite radius. It's a lot harder to fit in one's mind a sphere of infinite radius.

If there is any mathematical reasoning that can be considered safe, then it's manipulation of finite strings of objects or symbols (whether concrete sticks on the ground, or abstract tokens)...if I really had to, I could fall back to extreme formalism by taking the theorem to be utterly uninterpreted, but a formula nevertheless to be used in mathematical reckoning. — TonesInDeepFreeze

I completely agree. I can't imagine a scenario where the work that mathematicians have been doing for decades using set theory could suddenly be worthless. Maybe some ideas would need to be re-evaulated (e.g. Axiom of Choice and the Banach–Tarski paradox) but by and large I am completely convinced that the vast majority of modern math would retain it's value even IF actual infinities were banished.

as actual infinities (rather infinitesimals) were banished from (mainstream) calculus and replaced with potential infinities (through limits) — keystone

No, limits use infinite sets. The standard axiomatization of analysis is ZFC. Ordinary modern analysis is decidedly infinitisitic. Maybe you're thinking of the banishment of infinitesimals?

I think it's quite easy to imagine a closed finite universe, for example a sphere of finite radius. It's a lot harder to fit in one's mind a sphere of infinite radius. — keystone

A sphere has infinitely many points in it.

And is there such a thing as a sphere with an infinite radius? If I'm not mistaken the radius of a sphere is a real number, right?

by and large I am completely convinced that the vast majority of modern math would retain it's value even IF actual infinities were banished. — keystone

We've come around full circle. Now circling for another orbit:

To axiomatize mathematics sufficient for the sciences, without infinite sets requires not just deleting the axiom of infinity from (Z\R)+CC (ie. Z without Regularity but with Countable Choice), but a very different system. There are systems without infinite sets, but I don't know how well they do or how easy they are to understand and use.

What was the 19th century analysis resolution to Zeno's paradox?
— keystone

Infinite summation: convergence of an infinite sequence to a limit. — TonesInDeepFreeze

I believe that, contrary to the mainstream interpretation, 19th century analysis is the mathematics of continua (not the mathematics of points). I do believe that the mathematics of continua (i.e. 19th century analysis) does indeed resolve Zeno's paradox. So in this partial sense I agree with you.

However, I don't believe that a point-based interpretation of 19th century analysis allows for Zeno's paradox to be resolved. If it does, then answer this: If I travel along a line from x=0 to x=1, what is the next point that I travel to? As you know, there is no 'next point' so Zeno would claim that it's impossible to take a first step. As long as we hold on to points Zeno's paradox stands.

Are you not disquieted that a subset of rooms is equinumerous to the full set of rooms?
— keystone

I don't conceive of an infinite set of physical rooms.

As to sets, I already mentioned that I am not bothered that the squares (a proper subset of the naturals) is 1-1 with the naturals. — TonesInDeepFreeze

It sounds like you don't think Hilbert's Hotel is a good analogy then. Is that true? I like discussing in terms of the hotel because it's clear that we're talking about an actually infinite object not a potentially infinite process. For example, I have no problem with programming a computer algorithm to print n:n^2 for all n. Such a program could be written in a few lines of code. My issue is with the notion that such a program could be executed in completion. The complete output of the program would be an actually infinite object which I don't believe in.

I don't think calculus needs actual infinity to work.
— keystone

It does in its common form. — TonesInDeepFreeze

Please provide the simplest example you can think of where it is needed? I'd like to see whether I can challenge it.

I suspect (with no evidence to provide) that ZFC doesn't need actual infinity to work either.
— keystone

It wouldn't be ZFC then. — TonesInDeepFreeze

Just as I can write a program that outputs all natural numbers, I think it is fair game to talk about an infinite set. Might it be possible that ZFC is useful mathematics that talks about things that don't actually exist? Might the useful thing not be the objects but the 'talking'. Might we be able to reinterpret the axiom of infinity as 'we can talk about infinite sets'?

The imaginary hotelier can do that also. — TonesInDeepFreeze

Please allow me to indulge in Hilbert's Hotel a little more just so that we can stick with actually infinite objects instead of potentially infinite processes. As such, I want to avoid resorting to algorithms or properties to ensure that we are clear on our differences. In the lobby there is a lamp that is currently off. Each room has a switch to invert the state of the lamp. When each guest is moved to the next room, the guest flips the switch. After infinite guests move rooms, is the lamp on or off? There's no way around it...Hilbert's imaginary universe (where infinite processes can be completed) doesn't make sense.

How is that substantively different from Thompson's lamp?

I already responded regarding Thompson's lamp.

/

I don't know a theorem of set theory that is rendered as "infinite processes can be completed".

Set theory doesn't axiomatize thought experiments.

potential vs actual infinity — apokrisis

The distinction makes sense if we approach numbers as entities generated by a process, an algorithm. An instruction set that generates the natural numbers (vide infra) will go on forever, it's a task that's endless ($\infty$).

1. n = 1
2. print n
3. n = n + 1
4. goto 2

An actual infinity would be impossible, algorithmically speaking because for that, the algorithm must terminate, but as you can see it's a bloody loop.

It's a non finito and perhaps that's the whole point.

Yes, with that algorithm, at no step is there generated an enumeration of all the natural numbers.

But that doesn't prove that there does not exist a set whose members are all and only the natural numbers or that there does not exist an infinite set.

And it doesn't prove even the weaker claim that it is not provable that there exists a set whose members are all and only the natural numbers or that it is not provable that there exists an infinite set.

The non sequitur is:

R does not provide us with W, therefore there is no T that provides us with W.

One might as well say, "A hammer won't lift a beam, therefore nothing will lift a beam".

If we want for it to be provable that there does not exist an infinite set, then we need axioms to do that.

If we delete the axiom of infinity from ZFC and add the negation of the axiom infinity, then we prove that there does not exist an infinite set. But that theory is inter-interpretable with first order PA, which does not provide a calculus for the sciences.

And if we merely want for it not to be provable that there does not exist an infinite set, then we merely need to delete the axiom of infinity from ZFC, and then we can't prove that there exists an infinite set and we can't prove that there does not exist an infinite set. And that theory does not provide a calculus for the sciences.

This is to say that it is fine to mention the well known ostensive illustration of the notion of 'potential infinity', but that's all that is - an ostensive illustration of a notion; it doesn't even hint at how we would make a theory from it.

If we want for it to be provable that there does not exist an infinite set, then we need axioms to do that. — TonesInDeepFreeze

You’ve got it back to front. It has been shown there is a generator that can produce an arbitrarily large number. You’ve accepted that. Now what needs proving is that there is also this set containing an infinity of numbers.

Why do you keep shifting the burden in your posts?

What I fail to understand is why is it necessary for at least one (I'm told this is {1, 2, 3, ... } , the set of natural numbers, probably the foot in the door) $\infty$ to exist for mathematics to work?

Does finitism mean some domains of math vanish into thin air?

(1) 'finitism' has different senses.

(2) Perhaps it is not necessary to have infinite sets for an axiomatization of mathematics for the sciences. It's just that in order to evaluate a non-infinitistc axiomatization, we need to have it specified.

I believe that irrationals are algorithms which describe this mysterious other object - continua.
— keystone

I've already raised this point, asking if sqrt(2) is the exception or the rule. Higher dimensional generators could produce generators of some number that looks to be an irrational point of the line. But then these numbers - growth constants like e and phi – are ratios and so are dimensionless unit 1 values more than they are some weird real number.

The status of any regular irrational seems different. They would lack generators apart from decimal expansion. Something else is going on. — apokrisis

Undefinable real numbers have no place in my view. Focusing solely on definable irrationals, I believe an irrational is irrational, no matter how efficient/beautifully it can be expressed. My point is the rule not the exception.

I don't believe there is a fundamental length since any length can be divided.
— keystone

Can the Planck length be divided? Not without curling up into a black hole.

What you believe and what the Universe would like to tell you seem two different things. Who wins? — apokrisis

It has not been determined whether space is discrete or continuous (LINK). I'm inclined to believe that the planck length is a limitation that is applied to measurement, not the divisibility of space itself.

Your trust in physics has made you believe something obviously counter to your mathematical intuitions - that any positive number can be divided.

I can imagine a mind that lives in a 4D universe that can picture a 4D hypersphere as easily as a sphere.
— keystone

Yea, nah. I'm not buying these feats of your imagination. — apokrisis

Seriously? My mind can only hold so many bits therefore there is a largest number that I can retain in my memory. However I can easily imagine a mind whose memory could retain a larger number. It's the same thing with dimensions. No leap of faith is required here. The real feat of imagination is imagining a mind whose memory could hold an infinitely large number. That I cannot do.

the one where the world is "thinking itself" into definite being in ontic structural fashion. — apokrisis

In my stereogram analogy I do not exist within the page, and neither does the computer exist within the continuum. Or perhaps more digestible, neither does the computer exist within the simulation.

The hidden rabbit or seagull is merely hidden while the brain finds a way to suppress the shapelessness of the coloured pixels from the intelligibility of a depth perception-based contour. — apokrisis

It sounds like you're saying that the rabbit exists even when not observed. I would argue that it only has the potential to exist and it actually exists when observed correctly. In any case, I think you're nitpicking my analogy. If I handed you a blank white piece of paper you could argue that it is a picture of a polar bear playing in the snow but I would argue that it only contains the potential to be such a picture, and it would actually be that picture only once you cut out the bear figure with scissors.

The page contains the potential of infinite images,
— keystone

The problem is that it doesn't. It plays on a dichotomous rivalry of brain subsystems. You have to switch off the one and employ the other. The search is for the single hidden interpretation. Only one of the two points of view can spot it. — apokrisis

Perhaps I'm missing your point. Do you agree that given a continuum there's infinite potential to how you cut it up?

This is why the brain is not a computer. — apokrisis

I would prefer to say 'this is why the brain operates different from a modern computer'. I agree with Max Tegmark when he said 'we should reject carbon-chauvinism'. However, this is already a broad discussion, I don't think we should extend it to debate this.

Yes, this will seem very counterintuitive. The simplest way I can explain it in a non-technical fashion is that selecting any non-zero probability for each number will force us to add up way over 100%, because there are infinitely many other "participants" (numbers), which means the only probability we can assign to each participant is zero. — Kuro

I understand how we got here, it's just that to me this screams that what got us here is a mistake.

There's actually a way out of this being nonstandard analysis — Kuro

I have a hard time accepting real numbers, I have an even harder time accepting hyperreal numbers.
But yes there is a very simple way out of this, and that is that points don't exist. When you throw a dart at a dartboard, you don't hit a point, you hit an area. Any discretization of a dartboard into areas produces a finite number of areas each with a finite probability, all summing to a probability of 100%. What's wrong with this view?

(1) 'finitism' has different senses. — TonesInDeepFreeze

:up:

(2) Perhaps it is not necessary to have infinite sets for an axiomatization of mathematics for the sciences. It's just that in order to evaluate a non-infinitistc axiomatization, we need to have it specified. — TonesInDeepFreeze

As I've argued, science is founded on the pragmatism of the semiotic modelling relation. That changes the view of the situation. Axiomatic deduction becomes just the middle part of a three stage process. The formality of the deductive step is matched by the informality of the initial abductive step (the inductive leap to a possible answer) and then the informality (or pragmatism) of the inductive confirmation – the act of measurement, the act of actually applying the idea of counting to the world which needn't necessarily be crisply countable.

You say the idea of the stick – as an atomistic object – gives safe ground for counting in terms of abstract objects. But are any two sticks ever alike outside your willingness to grant them the status of being "sticks" of near enough similarity in terms of size, shape, weight, etc, to ignore the differences and declare them "the same"?

So the idea of counting is different from the physical act of counting. One deals in abstract objects, the other in real objects. And all real objects full of contingency. They have any number of defects and blemishes. Indeed, the ontology of real objects – for the Peircean pragmatist – is that they are simply a state of contingency that has been constrained to some suitable degree where the differences statistically cease to matter. The set of physical objects is constructed as the range under the generator of some agreed distribution curve. Near enough becomes exact enough – in a way that is outside of deductive reason, but within grasp of the feedback loop of a hypothesis-testing pragmatism.

It is great that maths tries to make itself as robust as possible by an axiomatic approach. The methods of deductive proof. This part of the modelling relation needs to be as watertight as it can be.

But then science is equally concerned with the business of measurement. It may be informal in the logical sense, but it has to be strict in the methodological sense.

Then we must have a robust approach to abduction as well. And here things become murky. How does one bottle true creative insight of the kind that is the leap to the model that is going to work and achieve its pragmatic goals?

There are clearly ways to cultivate or train folk in abductive reasoning. But most education systems are more concerned to bash you over the head with a textbook.

Given your concern for the proliferation of cranks obsessed with offering original solutions to the biggest problems, the secrets of abductive thought should give clues to what goes wrong there.

Largely of course it is a failure to be iterative. To guess small and test. Then guess larger and larger in conjuction with broader and broader reality testing of the formal theory being constructed.

I can draw a line with open ends on a piece of paper and label the ends negative and positive infinity. This unbounded object is entirely finite.
— keystone

You can draw a sign that you then interpret in a certain habitual fashion. The issue then is how does this sign relate you to the reality beyond. Does is create a secure bridge? Or is it wildly misleading? — apokrisis

ELI5.

If maths has been left behind in this grand and still unfolding adventure, tough shit. — apokrisis

You believe that math itself has some fundamental limits, perhaps a frequency, a duration, or a length. You may be right but I think you're wrong in applying it to continua. Instead, you should include the computer as a fundamental participant in mathematics and apply these limitations to the computer. Apply them to measurement. It makes sense, after all. Every computer has its limits. It has a finite memory so there is a limit to the size of the numbers that it can store. It also operates at some frequency. This is all common sense and it requires no philosophizing. Mathematical objects don't exist eternally in the Platonic realm. They exist when we (computers) compute!

In set theory, there is no completeness axiom. Rather, we prove as a theorem that the system of reals is a complete ordered field. — TonesInDeepFreeze

Must an ordered field necessarily be a field of numbers? Could it instead be a field of equations? For example, I would imagine that adding a 1 to each number in the field does not negate the field.

And yes one might want for the axioms to be intuitively correct ("true") even if the theorems might be surprising. And with set theory, people's mileages vary. I find the axioms of set theory to be exemplary in sticking to only principles that are in concordance with the intuitive notion of 'sets'. — TonesInDeepFreeze

I largely agree with you about axioms. My only issue is that the axioms of set theory are not in concordance with the intuitive notions of 'finite sets'. And since the only sets we ever work with directly are finite, I think we should be cautious accepting axioms that oppose them. But as I mentioned before, a small tweak to the language of the axioms will be consistent with my intuitions. Instead of saying 'there exists an infinite set' I would be comfortable saying 'there exists an algorithm that describes an infinite set'.

Does finitism mean some domains of math vanish into thin air?

I am triggered by this. If you start to think anything and than stop, where does it go? If we think of an infinite line, it seems to have to vanish as one's mind caters to finite things. Not just this fact, but the fact that we do not have infinite life-spans also makes it impossible to somehow span one's consciousness in infinite directions. Birth and death create this finite life, and probably all forces of life thereof.

Undefinable real numbers have no place in my view. — keystone

You reject vagueness then. That is certainly the usual thing to do. :wink:

My point is the rule not the exception. — keystone

And how do you know there is a rule unless you have ever seen some exception? How can you define a definite limit unless you actually have indeterminate challenge of the borderline cases?

A rule is meant to range as a constraint over all possible exceptions. In practice, we live by rules that simply limit exceptionality to that which doesn't matter. We reign in the differences that matter and ignore the differences that don't.

I prefer the practical approach to these things rather than the Platonic. The advantage of indeterminacy is that it then gives you something to determine. There is a reason to have been thinking deeply about what matters and thus a choice that has been made which constructs a world with some actual meaning to it.

It has not been determined whether space is discrete or continuous (LINK). I'm inclined to believe that the planck length is a limitation that is applied to measurement, not the divisibility of space itself. — keystone

But the Plank scale measures the fundamental grain of indeterminacy – quantum uncertainty or vagueness. Determination can start from that point where the symmetry is dialectically broken. We can start to say that some particle has both a limited location and a limited momentum in some strong sense.

The limit of measurement is indeed vagueness or indeterminacy – the state where the PNC finally fails to divide the world distinctly.

It sounds like you're saying that the rabbit exists even when not observed. — keystone

But the image is constructed by having the contour information split into a pair of representations which you then have to fuse bak into one by changing your depth of view. So you could only ever see the suggestive contour of whatever object has been plugged in.

If I handed you a blank white piece of paper you could argue that it is a picture of a polar bear playing in the snow but I would argue that it only contains the potential to be such a picture, and it would actually be that picture only once you cut out the bear figure with scissors. — keystone

That is a better analogy. I prefer my own still – the static on the TV screen which is both every show you could ever see, but all at once ... or else just meaningless noise.

Perhaps I'm missing your point. Do you agree that given a continuum there's infinite potential to how you cut it up? — keystone

The mathematical picture of the continuum is of something that can be unlimitedly cut. But physics tells us that the real world works with cut-offs.

That doesn't mean there is some smallest interval – a point where the line and its gaps are the same size and so you can land cleanly either, and with equal probability, on the line or the gap. It means that there is some scale where the very distinction between lines and gaps becomes vague. It ceases to be a meaningful distinction.

If you agree with a cut based approach to the continuum, then this is what lies at the end of the trail. The cuts must have some size to actually disconnect the line at a point. The more cuts you make, the more the continuous line becomes a tissue of cuts. When you get down to the scale where the line segments and the cut widths are the same, it is 50/50 whether you have one or the other. And the next cut - in a single step - then removes the line completely.

So there are some issues if you want to claim bivalence all the way down to the limit.

I agree with Max Tegmark when he said 'we should reject carbon-chauvinism'. — keystone

What does Tegmark know about carbon then?

Biologists learn that carbon chemistry is unique because of things such as that it forms 29,000 hydrogen compounds to nitrogen’s 65, oxygen’s 21, or silicon's 55. And carbon makes metastable monomer chains where silicon only makes crystals.

In other words, if the physical world wanted to do "computing" in the style of genetic codes providing the constraints to organise chemical potentials to do work, carbon was the only thing that made it a true possibility.

Rather than reject carbon-chauvinism, we ought to reject basic biological ignorance.

Must an ordered field necessarily be a field of numbers? — keystone

No. But all complete ordered fields are isomorphic with one another. So all complete ordered fields are isomorphic with the system of reals.

Could it instead be a field of equations? — keystone

I know there are field equations (but I know really nothing about them), but I don't know what a field of equations is.

the axioms of set theory are not in concordance with the intuitive notions of 'finite sets' — keystone

All the axioms are in that concordance, except one.

And since the only sets we ever work with directly are finite, I think we should be cautious accepting axioms that oppose them. — keystone

I think there's something to that. Indeed, it is, roughly speaking, right in line with Hilbertian finitism. Hilbert's idea was that we can work in infinitistic mathematics if we have a finitistic proof of the consistency of infinitistic mathematics. Famously, we found out that there is no finitistic proof of the kinds of systems we'd like to use, not only not of set theory but even of PRA, the system itself that we may take as exemplifying finitistic reasoning at its "safest". Yet, if I understand correctly, Hilbert's condition was a sufficient condition not a necessary one.

Instead of saying 'there exists an infinite set' I would be comfortable saying 'there exists an algorithm that describes an infinite set'. — keystone

That's interesting. But, if that is to be a statement in the system, we'd need to see "described" couched mathematically. I have a hunch that your notion is pretty much the same as 'there exist potentially infinite sets', and as I've said, I don't know a system that says it.

My argument is that the whole potential vs actual infinity thing comes from the fact that our ideas about numbers are based on systems of constraints. — apokrisis

I disagree. I think that potential infinity comes naturally to us, whereas we reluctantly hold on to actual infinity. We do so because we are not used to the notion of the observer (mathematician/computer) playing an active role in mathematics. As with classical physics, we want the observer to play a passive role by making passive observations upon the objects of math that exist in completion for eternity in the Platonic realm.

But you're a fan of physics so you should appreciate that in QM the observer plays an active role converting the potential to the actual (by means of making measurements to collapse the wave function). Consider the possibility that mathematics is contingent upon the interaction between the computer and the continuum (by means of cuts), in the same way that reality is contingent upon the interaction between the observer and the wave function (by means of measurements).

How many numbers are there?
A mathematical platonist would have to say infinite. I would go in a different direction. I would ask 'where?' How many numbers are 'where'? In other words, in what computer/mind are you talking about. You have to be specific about where is because there is no Platonic realm. And right now, as I'm getting ready to go to bed, if you ask me 'how many number are there in my mind?' I would answer zero.

Well, take Cantor as an example - while most others were loathe to get involved with $\infty$, he leapt at the opportunity and what do you know?, the answer to the question "what is $\infty$?" lay in an ancient method for counting, the one-to-one correspondence (unique pairing of items in one group with that of another). That's like solving an engineering problem in spaceflight using a ploughshare, oui?

We need another Cantor to make the next leap in our understanding of $\infty$. I hope he's out there somewhere. Fingers crossed.

My interest in finitism hearkens back to the Greeks I suppose. $\infty$ defied common sense intuitions e.g. an infinite set and its proper subset have the same cardinality which in colloquial terms means a part is equal to the whole. Some might disagree on this point, but there are peeps who say this is exactly what the mathematics says. $\infty$ maybe the internal combustion engine of math - creates more problems than solutions.

No, limits use infinite sets. The standard axiomatization of analysis is ZFC. Ordinary modern analysis is decidedly infinitisitic. Maybe you're thinking of the banishment of infinitesimals? — TonesInDeepFreeze

I can't speak to the standard axiomatization of analysis, but the informal definitions that us engineers were taught didn't use sets. As written on Wikepedia: a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. I take "approaches" to be a potentially infinite process.

an infinite set and its proper subset have the same cardinality — Agent Smith

Just to be clear, an infinite set has the same cardinality as some of its proper subsets, but not all of them.

in colloquial terms means a part is equal to the whole — Agent Smith

No it does not. It means a proper part is in 1-1 correspondence with the whole, not equal to the whole. A proper subset S of T is never equal to T. I've mentioned this before, but you repeat your disinformation.

there are peeps who say this is exactly what the mathematics says — Agent Smith

Who specifically?

∞ maybe the internal combustion engine of math - creates more problems than solutions. — Agent Smith

I can only guess at what vague thing you mean every time you write the leminscate.

I don't know what problem, especially a practical one, you think is caused by set theory.

You believe that math itself has some fundamental limits, perhaps a frequency, a duration, or a length. — keystone

That isn't my argument. Maths can be treated as its own abstract game with its own generating axioms. That game generates a bunch of abstract nonsense, but also occasionally hits on something "unreasonably effective" so far as modelling realty is concerned.

The issue is with the abductive end of the story – the intuitions that might guide the further progress of our reality modelling as a community of philosophers, scientists and mathematicians.

If your intuitions are wedded to the concrete machinations of deductive algorithms, then you might be locking yourself into a world picture where you can't see it as instead a story of self-organising emergence with natural cut-offs.

Finitism indeed might not be the best way to extend the reach of the formalisable. But it may instead be the best way to reconnect the formalist project to the reality it left behind.

That is the debate I am focused on.

Every computer has its limits. It has a finite memory so there is a limit to the size of the numbers that it can store. — keystone

But I've already said this about computers. The abstract formalism – the Turing machine – has an infinite length of tape and infinite time to move it back and forwards through the single digital gate. However real computers live with real physical restrictions and their maximum information capacity is constrained by the Bekenstein bound. This is the point at which the computer turns into a black hole under its own gravitational pull.

So yes, we have theories about how to distinguish between the mathematical fictions and the real world realities. Time and energy become constraints on counting or cutting – as the intuitionists argued.

Mathematical objects don't exist eternally in the Platonic realm. They exist when we (computers) compute! — keystone

And yet Universal Turing Computation is a mathematical object – conceived in Platonia. This is the kind of "paradox" we are meant to be figuring out here, not simply saying one is the other as if the differences were moot.

A sphere has infinitely many points in it.

And is there such a thing as a sphere with an infinite radius? If I'm not mistaken the radius of a sphere is a real number, right? — TonesInDeepFreeze

Let's drop a dimension for simplicity. I can think of x^2+y^2=1 without having to think of any points. I can also draw that function without drawing any points (I can show you if you're curious). The points come from the circle, not the other way around.

No, there is no such thing as a sphere with infinite radius...that's explains why I can't imagine it.

the informal definitions that us engineers were taught didn't use sets. — keystone

Mathematics, in many branches, is brimming with sets. Analysis, topology, abstract algebra, probability, game theory... Can't even talk about them, can't get past page 10 in a textbook, without sets.

But of course, one can use the theorems of mathematics for engineering without tracing the proof of those theorems back to axioms, in particular the set theoretic axioms. That's not at issue.

a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value — keystone

A sequence is a set. And it has a domain, which is a set, and a range, which is a set. An infinite sequence is an infinite set with an infinite domain.

Of course, one can leave that unconsidered, not in mind, when working in certain parts of calculus. That is not at issue. But when we trace the proofs of the theorems of analysis back to axioms, then, in ordinary treatments, those are the axioms of set theory.

I can think of x^2+y^2=1 without having to think of any points — keystone

Of course. No one says that you have to.

The points come from the circle, not the other way around. — keystone

A circle is a certain kind of set of points. I don't know what you mean.

No, there is no such thing as a sphere with infinite radius...that's explains why I can't imagine it. — keystone

Yes, and since infinitistic mathematics doesn't have spheres with infinite radi, infinitistic mathematics doesn't call on you to imagine it.