Thread: Ultrafinitism

let me say at the outset, that i think the topological concepts enabled by the real number system are in their own right, beautiful and elegant. and the effort put into a rigorous definiton of, say complex analysis, has certainly paid off in many pragmatic applications. that said, what evidence do we have that non-finitary methods are the most "realistic" in terms of what we can deduce about the world?

proof by induction, is a perfect case in point. i know the method, as well as its set-theoretic justification. but if we were to drop the axiom of infinity from ZF, the validity of the induction axiom schema of peano arithmetic seems questionable.

i guess what i want to know is, what concrete reason do we have, for supposing that Q models anything actually in existence, rather than say Zp, for some extremely large prime p? i realize the enormous implications of this: most "free" objects (functions spaces, free groups, etc.) go away, their existence becomes "purely hypothetical" (although perhaps one might say they already are). i'm not selling a concept, here, i just would like to know people's opinions. is this something anyone even cares about?

Ultrafinitism is a solution to the foundation crisis that has no interest at all. If we admit it, we might as well stop doing mathematics.

well, obviously, you're not interested in it, and that's cool. but i'm kind of curious about it. it does have it's detractors, and i can understand why you're averse to it.

i see no reason not to investigate differing philosophies, and i don't feel bound to subscribe to something just because i know about it.

I didn't mean to be rude regarding your interest in exploring ultrafinitism. If you're interested in exploring various philosophies, that's great.

i understand that ultrafinitism pretty much guts the real number system, and many of the more interesting topological structures, large swaths of set theory, etc.

what i want to know, particularly, is if we regard mathematics as a useful tool in modelling the world we live in, how do we a priori decide that non-finitary structures are more accurate?

obviously, the decision to go this route was taken quite some time ago, and adherents of finitism in mild or industrial strength are not in favor these days. but why this should be so, is not immediately apparent to me, especially since the advent of quatum physics, which posits that there is a mesh size that effectively limits the size of the knowable universe to us (at least on scales of smallness. at the other end, apparently the relativistic constant c, limits the size of the knowable universe at the other end, although there is no reason to suppose there isn't more "beyond").

a forum i used to frequent (not a math or science one) had a long on-going debate on whether the nature of our world was digital (discrete) or analog (continuous). personally, i favor a "fractal" view of the world, there seems to be a similar level of complexity on all scales of observation, which argues (persuasively, for me) an "infinte" view. but, it could be that the number of iterations is just higher than we can account for.

also, i am curious in what people believe, and why they believe it. sometimes the reasons people give for their beliefs are powerful and persuasive, and influence my thinking for long afterwards.

Originally Posted by Deveno

i understand that ultrafinitism pretty much guts the real number system, and many of the more interesting topological structures, large swaths of set theory, etc.

what i want to know, particularly, is if we regard mathematics as a useful tool in modelling the world we live in, how do we a priori decide that non-finitary structures are more accurate?

You know nothing a-priori about what mathematical structures are appropriate for physics, you find out by trying them. There is no reason why the mathematics we use should correspond to the fundamental nature of reality, we are interested in them of themselves, and apply them where/if they work.

snip ... which posits that there is a mesh size that effectively limits the size of the knowable universe to us (at least on scales of smallness. at the other end, apparently the relativistic constant c, limits the size of the knowable universe at the other end, although there is no reason to suppose there isn't more "beyond").

You and/or your source are just making this up, we know no such thing/s (it may be an interesting line of inquiry but there is a difference between what we know and speculative proposals)

CB

Originally Posted by CaptainBlack

You know nothing a-priori about what mathematical structures are appropriate for physics, you find out by trying them. There is no reason why the mathematics we use should correspond to the fundamental nature of reality, we are interested in them of themselves, and apply them where/if they work.



You and/or your source are just making this up, we know no such thing/s (it may be an interesting line of inquiry but there is a difference between what we know and speculative proposals)

CB

would you say, then, that mathematics is "fiction", albeit chosen by some sort of "best-fit" criterion?

it is my understanding (and i may be wrong, or perhaps merely misinformed), that at distances less than 1 planck length, quantum effects dominate, and the hesisenberg uncertainty principle meakes measurement impossible. moreover, in string theory, there simply aren't any shorter lengths. i'm no physicist, but apparently many people actually believe that this is the case; some even think that space-time is discrete (or perhaps, foamy-depends on who you talk to) at that scale.

at the other end, events outside the "light cone" (also known as "the absolute elsewhere"), may occur, but there is no way for information from these events to reach us.

i appreciate that these are just "theories", but...so is electricity, right? i mean. it is not my intention to get into the can of worms that is: "do we know anything at all? and if so, how do we know that we know?" and beyond that, "how do I know that what YOU know, is the same knowledge that I know, even when we talk to each other, in the same language?" it's not that these aren't interesting questions, in their own right. just a bit afield of my original musings.

in a sense, all science is speculative. there are reasons to prefer some speculations over others (a preponderence of evidence).

i'm all for "doing mathematics for its own sake". but there is a strong thread of justification of mathematics according to its applications running through most types of science/engineering education, for example. and people get all existential about the real number system, like it's not just something we made up.

Originally Posted by Deveno

would you say, then, that mathematics is "fiction", albeit chosen by some sort of "best-fit" criterion?

it is my understanding (and i may be wrong, or perhaps merely misinformed), that at distances less than 1 planck length, quantum effects dominate, and the hesisenberg uncertainty principle meakes measurement impossible. moreover, in string theory, there simply aren't any shorter lengths. i'm no physicist, but apparently many people actually believe that this is the case; some even think that space-time is discrete (or perhaps, foamy-depends on who you talk to) at that scale.

You scattering of this with "many people believe" proves my point, many believing is not synonymous with knowing. Your faith in QM (or string theory) being a valid model in energy regimes so remote from what we can investigate is charming but probably misguided. Where experiment has not been (or cannot at present go) all is speculation. Now making predictions from whatever theory is flavour of the month is good, it gives us an idea where to look and what experiments might produce interesting results.

at the other end, events outside the "light cone" (also known as "the absolute elsewhere"), may occur, but there is no way for information from these events to reach us.

So what?

i appreciate that these are just "theories", but...so is electricity, right? i mean. it is not my intention to get into the can of worms that is: "do we know anything at all? and if so, how do we know that we know?" and beyond that, "how do I know that what YOU know, is the same knowledge that I know, even when we talk to each other, in the same language?" it's not that these aren't interesting questions, in their own right. just a bit afield of my original musings.

in a sense, all science is speculative. there are reasons to prefer some speculations over others (a preponderence of evidence).

You seem to have a problem distinguishing between things we can investigate experimentally and things that we cannot at present investigate experimentally

i'm all for "doing mathematics for its own sake". but there is a strong thread of justification of mathematics according to its applications running through most types of science/engineering education, for example. and people get all existential about the real number system, like it's not just something we made up.

Actually the real number system is something we made up, but not without reason. It is a mathematical construct which has properties that represent apects of measurement processes, but is an idealisation of such processes. There are other constructions of real number-oid systems which lack certain features (or indeed have additional features) of the usual real number system but they are not popular in engineering and physica because they are either more cumbersome to use or lack some property that we rather like.

CB

Originally Posted by CaptainBlack

You scattering of this with "many people believe" proves my point, many believing is not synonymous with knowing. Your faith in QM (or string theory) being a valid model in energy regimes so remote from what we can investigate is charming but probably misguided. Where experiment has not been (or cannot at present go) all is speculation. Now making predictions from whatever theory is flavour of the month is good, it gives us an idea where to look and what experiments might produce interesting results.

i have no faith in any of these theories. my point was, and continues to be, that some theories now entertained by some people, suggest a finitary model. it matters not to me if this is "true", or not. if what we conjecture (at present, and i use the term "we" loosely) suggests this, it is frankly a mystery to me why the infinite is so entrenched in mathematics. i do not question, say, the aesthetic value of such an idealization, i wonder why we think infinite structures are somehow so "natural".

So what?

well, if the universe is bounded, for example, it casts some doubt on the axiom of infinity.

You seem to have a problem distinguishing between things we can investigate experimentally and things that we cannot at present investigate experimentally

it is my understanding that often the theory pre-dates the verification. yes, experimental verification is vital, but what do we look to, to verify? it seems circular reasoning to only investigate the things we already know, hmm?

Actually the real number system is something we made up, but not without reason. It is a mathematical construct which has properties that represent apects of measurement processes, but is an idealisation of such processes. There are other constructions of real number-oid systems which lack certain features (or indeed have additional features) of the usual real number system but they are not popular in engineering and physica because they are either more cumbersome to use or lack some property that we rather like.

CB

a partial enumeration would be helpful, here, if only to satisfy my curiousity as to which number systems you mean. popularity of a number system means nothing to me, the majority view of a society is usually uninteresting to me (and math by democracy is a terrible idea).

I have apples, and someone gives me another apple. So I have...what, zero apples? Gah!

(Of course, I am assuming that they are assuming that the is so large this could never happen...but still...)

I'm very far from knowing anything substantial about physics. I'm not much closer to knowing something substantial about the foundations of mathematics, indeed I not very close to knowing something substantial about mathematics I think. But I still want to post here because this question has been an interesting one for me for a long time. I know I'm exposing myself to ridicule, but never mind that.

What I think now is that assuming the existence of uncountable sets is the lesser evil. It's just a feeling, not a well-grounded opinion, but I think problems with the axiom of choice (which would be avoided by giving up infinities) for example are irrelevant for our understanding of the real world. Same goes for classical counterexamples in real analysis. I feel they are all just a flaw of the system and that they shouldn't be there. And I think some mathematicians share my point of view -- the prefix "counter-" might be a clue. But then, I know that physicists do use continua for modelling the real world and consider it beyond their capabilities to model the real wold without them. I don't really see the reason very clearly but this is what I read. And it seems understandable even for me, a layman. It seems awful not to be able to measure a circle's circumference. Or just not having circles at all.

As for countable sets, I think they're just fine. Even if there's nothing infinite in the world (whatever this means), cutting off the natural numbers larger than some large natural number seems an unnecessary complication to me. Why bother? I don't know any result concerning natural numbers that would trouble my intuition... Well, the countability of rationals did at first but you can get used to it quite easily. Does anyone have an intuition justifying the Banach-Tarski decomposition? Has it ever been considered in physics?

Originally Posted by ymar

I'm very far from knowing anything substantial about physics. I'm not much closer to knowing something substantial about the foundations of mathematics, indeed I not very close to knowing something substantial about mathematics I think. But I still want to post here because this question has been an interesting one for me for a long time. I know I'm exposing myself to ridicule, but never mind that.

What I think now is that assuming the existence of uncountable sets is the lesser evil. It's just a feeling, not a well-grounded opinion, but I think problems with the axiom of choice (which would be avoided by giving up infinities) for example are irrelevant for our understanding of the real world. Same goes for classical counterexamples in real analysis. I feel they are all just a flaw of the system and that they shouldn't be there. And I think some mathematicians share my point of view -- the prefix "counter-" might be a clue. But then, I know that physicists do use continua for modelling the real world and consider it beyond their capabilities to model the real wold without them. I don't really see the reason very clearly but this is what I read. And it seems understandable even for me, a layman. It seems awful not to be able to measure a circle's circumference. Or just not having circles at all.

As for countable sets, I think they're just fine. Even if there's nothing infinite in the world (whatever this means), cutting off the natural numbers larger than some large natural number seems an unnecessary complication to me. Why bother? I don't know any result concerning natural numbers that would trouble my intuition... Well, the countability of rationals did at first but you can get used to it quite easily. Does anyone have an intuition justifying the Banach-Tarski decomposition? Has it ever been considered in physics?

as Cap'n Black pointed out, the usefulness of infinite systems is their "ideal" behavior. we don't have to worry about hitting some hypothetical ceiling, we can always apply a limiting process to get something that for lack of a better word i'll call a number. topological completeness (which for the countable base of rationals implies uncountability) is nice, because then cauchy IS convergent (sequences behave well).

take set theory, for example. we seem to be very hungry for a large collection of sets. rather than prove things number-by-number for arithmetic facts, it seems more intellectually satisfying to prove something "all at once" for EVERY number, which then settles the question for once and for all. and as soon as we have the capability of forming power sets, and we have an infinite set, boom! we have an uncountable set. and then strongly inaccessible cardinals, and so on and so forth. the level of abstraction gets raised so high, that whatever relationship it might have had to ordinary reality is tenuous. mathematics has become a world unto itself.

and your point is well-taken, why bother? we've built up a large body of knowledge which is beautiful AND useful, so why fix it, if it ain't broke?

i guess the way i see it is this: i am trying to imagine what a universe without infinity might BE like. how much smaller is what we know, and how much of what we've been calling "true", can we save? i mean, if the integers are really Zp for some very very large p, how does that change how we define things? i'm not going to stop using induction in the near future, or pretend i never heard about the real numbers, or complete metric spaces, or hilbert spaces, or any of that. i'm thinking along the lines of "an alternate mathematics", much like non-euclidean geometry is "an alternate geometry" (or two or three, or dare i say it? infinity).

Originally Posted by Swlabr

I have apples, and someone gives me another apple. So I have...what, zero apples? Gah!

(Of course, I am assuming that they are assuming that the is so large this could never happen...but still...)

well, that's just it. there wouldn't BE another apple to add. or, more accurately, the universe is made of up some number of "somethings" where just one more would completely destroy its internal structure. there wouldn't be any ROOM. for this very large number N, it would act like an ideal. and yes, i know this would pretty much make your life's passion pointless (being the infinite group lover that you are).

i think the idea of "dedekind-infinite" is a really good one (this is where all those lovely "hilbert hotel" tales come from). and taking that as the definition of infinity, is equipotent to the axiom of choice. i like the axiom of choice, i feel it should be "true" (just pick one!), but the fact that we can't prove it (we have to make it an axiom), puts it on the same sort of footing as the axiom of infinity. if n+1 (ok, for you purists out there, if x-->x U {x} is an injection) HAS to be different than every other k (including k = n), we're forced into infinity. i guess the 5-year-old in me is still asking: "why"?

Originally Posted by Deveno

i think the idea of "dedekind-infinite" is a really good one (this is where all those lovely "hilbert hotel" tales come from). and taking that as the definition of infinity, is equipotent to the axiom of choice. i like the axiom of choice, i feel it should be "true" (just pick one!), but the fact that we can't prove it (we have to make it an axiom), puts it on the same sort of footing as the axiom of infinity. if n+1 (ok, for you purists out there, if x-->x U {x} is an injection) HAS to be different than every other k (including k = n), we're forced into infinity. i guess the 5-year-old in me is still asking: "why"?

The way I view infinity is as a definition. Whether it "exists" or not is a moot point - we define it, and then we study this definition, much like the rest of mathematics! Of course, there are a number of definitions of "infinity" (cardinal, ordinal, etc.), and so you pick the definition which is relevant to what you are doing.

Of course, this isn't a philisophical point of view, but it means I can study infinite groups and stuff and not care about what ultrafinitists and their ilk think!

well i think category theory is nifty (i wish i understood it better), and of course, categories are even bigger ("more infinite") than sets (most categories of interest are at least proper classes, and the category Cat is...well, i don't even know what it is. an ensemble, maybe?). so i certainly think that if "infinity didn't exist, it would be necessary for us to invent it" (to badly paraphrase voltaire).

but i have these nagging doubts about how well-founded our assumptions are. have you ever heard of max tegmark? i saw an interview with him once, where he said:

"well, on mondays, wednesdays and fridays i think the universe is infinite, and on the other days i think the universe is finite."

and he's pretty out-there...his view is that the universe literally IS a mathematical structure, and that we might be able to even decide one day WHICH structure. he teaches at MIT, so it's not like he doesn't have solid credentials. here is his paper, which makes for interesting reading:

[0704.0646v2] The Mathematical Universe

to be honest, saying that we study infinite structures "because we can" is a perfectly valid reason to do so. part of what i'm interested in, is the flip side of that coin. ok, we study groups, right? and groups have certain properties that we can "leverage" to squeeze out information just from "groupness" without delving into "which group" we have. so one natural distinction that comes up is abelian versus non-abelian. or free versus defined by relations. we get different classes of results by changing our assumptions. we know pretty well what most of the logical consequences of ZFC are (which is why we USE it). what do we get, if we remove some of the axioms? (just like studying semi-groups, say, instead of groups).

to go back to the euclidean/non-euclidean analogy...just because non-euclidean models cropped up, did not mean we ditched euclidean geometry. on the contrary, we still use it as an essential ingredient in differential geometry. i'm interested in how far with can we go with finite methods, and what kind of landscape can we create with fewer tools? can we still formulate, say, godel's incompleteness theorem in such a setting?

Originally Posted by Deveno

well i think category theory is nifty (i wish i understood it better), and of course, categories are even bigger ("more infinite") than sets (most categories of interest are at least proper classes, and the category Cat is...well, i don't even know what it is. an ensemble, maybe?).

I don't know much about this but I'm pretty sure these are all just proper classes. What's an ensemble? Aren't classes just properties?