The debate was about inifinite vs. finite, not about a particular number. It was couched in the terms of angels rather than physical items specificaly to avoid preconceived notions about our physical world - notions which were limited, and recognized as limited, back then. Do we even recognize our notions as limited in our beautiful early XXI century...?
OP didn't argue that finite numbers are physical objects, they said that infinities are not present in the universe. For example, I could in theory hand you 7 electrons but there are not infinity electrons for me to hand to you.
Under some interpretations of physics there is a finite number of objects in the universe, hence a largest possible cardinality of things that is finite. Even sets like the number of possible states of the Universe appear to me to be finite, though so large as to be intractable to my imagination.
A conceptual circle and a real circle differ markedly because of the lumpiness of matter, quantisation, uncertainty. Every real circle is an approximation (yay Plato!).
Pi doesn't appear to me to be real, as in writ in matter/energy -- it's a ratio of measurements of an imaginary article, circles don't exist in our experiential 4-space.
Irrational numbers like root-of-2 just fall out of basic maths though, there's not going to be a real line with that measure just as there's no line with a predefined unit measure (ie unless you define that line at that point in time as your unit, but a real line isn't even straight).
I'd be really interested in refutation of this -- it's one of those positions I've held since Uni, at least, but never explored as I've never seen it challenged/supported nor named.
> But, at most countably many of those sets will ever be individually thought about by any human being.
If your arguments are tied to such physical constraints, then there aren't even countably infinite natural numbers unless the Universe is infinite in space or in time (such that there can be infinite humans or "thinkers").
I remember Dr. Zeillberger from my graduate days at Rutgers. He clearly had a lot of interesting stuff to say which he packages in sensational and emotional form. I found it off-putting then as well as now.
But to be charitable, I think it comes from a place of pushing with tremendous force a pendulum of thought back from the axiomatic nature towards more exploratory and relevant mathematics. It reminds me of Howard Zinn's intro to A People's History of the US in which he says that he purposefully did not water down his arguments to be more neutral because of the overwhelming nature of the opposition.
The infinity issue is quite interesting to ponder. Obviously with our current understanding of the universe and humans, the number of numbers ever encountered by all humans over all history would seem to be finite regardless of the fundamental nature of the universe itself. There is a very strong notion of our world being discrete in that way.
And it is very interesting to pursue these ideas. For example, the proof of infinitely many integers is to assume there is a largest and we add 1. But what if the largest is unknown as is the case much of the time? This is true, for example, with computers (it can be known for the native format, but that can obviously be extended in programming in an arbitrary way)
It is a practical question to ask about existence questions that require an axiom (axiom of choice often) which gives no constructive way of doing something. To what extent is that actually useful? And if we had an approximate construction of something without the axiom, but the thing itself may or may not exist depending on the axiom chosen, what would that mean?
It is also interesting to think about doing away with infinity and thinking about how annoying it would be to not be able to talk about pi or e or even sqrt(2). There would be no irrational numbers, presumably, in this framework.
One can then see the wonders that embracing infinity can lead to as a crutch for dealing with the horrendously messy world of the finite and discrete.
Discussing these issues when learning mathematics strikes me as elevating a great deal of bland rule absorption and gives students power over the tools of mathematics. So this debate is useful in my opinion though it would be absurd to try and draw a conclusion.
One last note about his discussion of the lack of rigor in QFT and that it was good. Recent work (at Rutgers[0]) suggests that at least some of the infinities and problems come from not appreciating the proper physical picture of what is actually happening. Nonetheless, the standard procedures worked to give us needed answers before such insight was discovered.
I think the basic answer should be that formal and exploratory mathematics are both useful and it is important to avoid dogma from either side.
I personally find the idea of finiteness much harder to accept.
Infinity makes sense to me. It just means there’s no limit to stuff. Why should reality have a limit?
A finite reality is just plain weird because it implies there’s a single actual number describing how much stuff there is. The universe contains X godzillion kilograms, or is Y billion light years across. Where did those numbers come from? What determined them? Why isn’t there more stuff, or less?
> There’s also nothing about the physical universe to suggest that it is finite, or that it cannot contain infinities. It’s simply an open question.
The only evidence we have is that it's finite. Just because the universe expands does not mean that at any given time it is not finite.
> What does the universe or its physical extent have to do with the nature of numbers?
The only way to have scientific confidence in an idea is to test it. We can't calculate infinity because, infinity being something that isn't finite, every unit of information in our universe would have to be used to describe infinity. This doesn't work because there is never infinite information space in our universe at any given slice of time. Think of it this way; you can't take a modern video game and get it to perform exactly the same on a home computer from 1996 because it simply lacks the computing capacity. The only way it can work is to reduce certain aspects of the software to make it work at a much lesser capacity. There are no examples of any system that can describe another system more complex than itself with total accuracy. Thus, it makes no sense that a universe in which we are currently only able to describe through finite numbers would be able to support calculating what infinity as well as support the rest of its contents, if it can even do that at all.
This is why it's not at all accurate that a set of numbers can be "infinite." It only seems infinite because, for all intents and purposes, we don't have the capability to keep dividing a range of numbers forever. Even if we tried, the inevitable dissipation of heat energy would prevent us from doing so, that is if we don't simply run out of finite resources before then. If there is something that is indeed numerically infinite, we too would have to be infinite in order to make sense of it. We can't actually do that. That would be a contradiction. To illustrate this, go write some code that calculates every single number that exists in the "infinity" between two numbers. You won't be able to. Your software will fail because your computer doesn't support it. That is unless it has infinite bits.
A set where there's "infinite" numbers would more accurately be described as being indeterminate. The seeming "infiniteness" of one of these sets breaks down when you realize there's no way to even demonstrate that part of a set is infinite. From a conceptual standpoint, it's similar to how it might seem that a ball will fall straight down when you drop it, and it's generally useful to think of gravity in such a way, but that doesn't mean that it's actually so, just as believing that a set can be "infinite" might be useful, but the use of the term "infinite" for something finite like a set is incorrect.
So no, to others who think I'm being off topic. This is entirely on topic. A set being "infinite" gives you the wrong idea. At best, it defines a vector too large for humans or even human computers to find an end to. It's entirely virtual until proven otherwise.
Infinity is not a number, it's a set of numbers, but specific infinities (e.g., the cardinality of the natural numbers, aleph-naught) are individual numbers, just not finite numbers.
[the divide] separates two kinds of mathematical statements: “finitistic” ones, which can be proved without invoking the concept of infinity, and “infinitistic” ones, which rest on the assumption — not evident in nature — that infinite objects exist.
I'm not saying there's a definition that works down that path (I don't know); I'm saying that "we only have multiple infinities because of this definition, which people objected to because it didn't give us enough infinities" seems a little muddled.
> This construction means there can't be an upper bound N because then step 2 couldn't be applied to N.
Bendegem discusses this problem at length in his paper [1]. As programming-heavy site, I assume we're all aware that computers have finite resources. The universe too has finite resources so no matter how big a computer you build, it too will be finite. Therefore the infinity that is so pervasive in math is unphysical in a very real sense. So what would math look like and how would theorems change if this finiteness were formalized? That's what various flavours of finitism aim to achieve.
So to get back to your question as to the nature of the naturals, it seems evident that yes, at some point, you literally can fail construct the natural number N+1 if you are given N, because you will run out of particles in the universe. What implications this will have for various theorems will be interesting for sure, but it isn't clear yet because finitism isn't given much funding.
Edit: however, it's clear that some very unintuitive results follow from the infinities embedded in mathematics, and that a finitist approach resolves some of them. For instance, the argument that "0.9999... = 1" is true in classical mathematics while this equality is arguably not true under strict finitism because "0.999..." does not exist, because infinite objects do not exist, and so it will never equal 1.
Well, if one assumes finiteness all of these paradoxes go away and this discussion is not about anything anymore. I studied physics and am quite attached to infinities. That one constrains oneself to a finite state system mostly does not mean that one is not allowed to look out of the window towards the stars and marvel at infinity anymore.
What, exactly, do you think I'm trying to argue? To me the question is exactly whether having a concept of infinity has metaphysical significance. I think no good answer is humanly possible. Your definition is just as tautological as anything offered in the positive. That's exactly the problem.
The debate was about inifinite vs. finite, not about a particular number. It was couched in the terms of angels rather than physical items specificaly to avoid preconceived notions about our physical world - notions which were limited, and recognized as limited, back then. Do we even recognize our notions as limited in our beautiful early XXI century...?
reply