Infinitesimals are deeply illogical/impossible concepts and are shunned by most of maths. — Devans99

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

The hyperreals are a model of the first-order axioms of the real numbers that contain infinitesimals. To construct them requires a weak form of the axiom of choice. In the 1970's nonstandard analysis, based on the hyperreals, was touted as a better way to teach calculus. Results are mixed. Most studies showed that students come out of standard or nonstandard calculus classes equally confused. No new math is introduced with the hyperreals; that is, anything you can prove with them you can already prove with the standard reals. That's why they haven't caught on. The standard reals have extensive mindshare. But infinitesimals have been made logically legit as of 1948.

If you truly want to understand the mathematics of infinitesimals, I recommend learning about synthetic differential geometry, also called smooth infinitesimal analysis. These links are to excellent brief introductions by Sergio Fabi and John Bell, respectively. — aletheist

Yes, yet another way to do infinitesimals. I'll check out your links since I don't know much about SIA.

A moment of time has a duration that is not zero, but is less than any assignable or measurable value relative to any arbitrarily chosen unit. — aletheist

Quibble I must. Nobody knows what a "moment of time" is, or if it's modeled by any of the various mathematical models of infinitesimals. I object to this claim. There's no currently accepted theory of physics that supports what you said AFAIK. Math $\neq$ Physics.

Bananas are a great source of potassium, and consequently, good laxatives as well.

I have come across some arguments that teaching calculus using SDG/SIA--grounded in category theory, rather than set theory--is more effective than either standard or non-standard approaches. In any case, scholars of Peirce's mathematical thought seem to agree that it comes much closer to being a rigorous implementation of his conceptions of infinitesimals and continuity than NSA. For one thing, the principle of excluded middle does not apply--i.e., the logic of SDG/SIA is intuitionistic, rather than classical--which is exactly what he consistently maintained about anything that is general, including anything that is truly continuous. For another, functions in SDG/SIA "are differentiable arbitrarily many times" (Bell), consistent with this statement.

Not only must any given instantaneous value, s, implied in the change be itself either absolutely unchanging or else always changing continuously, but also, denoting an instant of time by t, so likewise must, in the language of the calculus, ds/dt, d^2s/dt^2, d^3s/dt^3, and so on endlessly, be, each and all of them, either absolutely unchanging or always changing continuously. — Peirce, R 300, 1908

In any case, scholars of Peirce's mathematical thought seem to agree that it comes much closer to being a rigorous implementation of his conceptions of infinitesimals and continuity than NSA. — aletheist

So just as I call modern constructivism Brouwers revenge, I can call SIA Peirce's revenge. This is very interesting. Do you happen to know how SIA relates to constructive math? They both deny excluded middle as I understand it.

either absolutely unchanging or always changing continuously. — Peirce, R 300, 1908

What's R 300? Where can I find Peirce talking about calculus? I don't suppose I could ask for a simple explanation of what this phrase means? Is it anything like the quotient of dy and dx being the derivative when dy and dx "become" zero but aren't actually zero, as Newton thought of it?

ps -- Why are there so many Peirceans on this forum? I never hear about him anywhere else but his ideas are incredibly interesting.

For another, functions in SDG/SIA "are differentiable arbitrarily many times" — aletheist

Aha. That's also true of the complex analytic functions in standard math. If a complex function (function from the complex to the complex numbers) is differentiable once, it's automatically infinitely differentiable. So complex differentiable functions are extremely well-behaved. Whereas real number functions (reals to reals) can be very wild, and differentiable only once, or twice, or some finite number of times before they are no longer differentiable. I wonder if this is related to SIA somehow.

So just as I call modern constructivism Brouwers revenge, I can call SIA Peirce's revenge. — fishfry

Heh, I like it! I am an engineer, not a mathematician, so I would welcome your thoughts on SDG/SIA--although they probably belong in a new thread.

Do you happen to know how SIA relates to constructive math? They both deny excluded middle as I understand it. — fishfry

Not really, but I suspect that they have different reasons for denying excluded middle. You might be interested in John Bell's book, The Continuous and the Infinitesimal in Mathematics and Philosophy. The first half covers the history, while the second half consists of chapters specifically on topology, category/topos theory, NSA, constructivism/intuitionism, and SDG/SIA. A new version just came out with an even longer title, but best I can tell the only significant change is the addition of several appendices on various topics.

What's R 300? Where can I find Peirce talking about calculus? — fishfry

R 300 means manuscript number 300 as cataloged by Richard S. Robin in the 1960s. That particular text is incomplete and largely unpublished, but there is a transcription online. Peirce does not actually talk about calculus much in it, and I honestly do not know where else in his voluminous writings he might have done so in any detail. You might find some leads in the Robin catalog, and then you can browse through images of the actual manuscripts.

I don't suppose I could ask for a simple explanation of what this phrase means? Is it anything like the quotient of dy and dx being the derivative when dy and dx "become" zero but aren't actually zero, as Newton thought of it? — fishfry

Here is what Peirce wrote right before the sentence that I quoted.

Accepting the common-sense notion [of time], then, I say that it conflicts with that to suppose that there is ever any discontinuity in change. That is to say, between any two instantaneous states there must be a lapse of time during which the change is continuous, not merely in that false continuity which the calculus recognizes but in a much stricter sense. — Peirce, R 300, 1908

Presumably the "false continuity" that he had in mind was that of Cantor. Does this help at all?

Why are there so many Peirceans on this forum? I never hear about him anywhere else but his ideas are incredibly interesting. — fishfry

I am not sure that there are really so many of us here, but I obviously agree with that last part. The problem is that Peirce never wrote a magnum opus spelling out his entire philosophical system, or for that matter any significant portion of it.

The Collected Papers (eight volumes) are arranged topically, rather than chronologically, with some rather misleading results because his thought definitely evolved over time. Unfortunately, the comprehensive critical edition being prepared by the Peirce Edition Project has been stuck for ten years after producing only eight volumes of thirtyish planned, primarily due to chronic lack of funding. The Essential Peirce (two volumes) is probably the most accessible compilation, but emphasizes general philosophy. Two specifically mathematical compilations--Carolyn Eisele's New Elements of Mathematics (four volumes in five) and Matthew E. Moore's Philosophy of Mathematics: Selected Writings--are probably more relevant to you.

You can learn something on this forum. A number of years ago I bought Robinson's book on NSA and got excited about it for a while, even contemplated teaching calculus that way, but put in on a shelf instead and moved on to other ideas. But I had not heard of SIA until joining this forum. I can't say I am very impressed with it, however, not really wanting all my functions to be continuous and all! And the defining characteristic involving squared infinitesimals seems just another strange notion one can avoid. :wink:

so I would welcome your thoughts on SDG/SIA — aletheist

I don't know enough. I know that it's categorical in flavor ... but then again so is differential geometry. I'll dispatch a clone to study up on SIA and another clone to read the collected works of Peirce ... uh oh I haven't got any clones. That means these tasks might not get done any time soon if ever.

Accepting the common-sense notion [of time], then, I say that it conflicts with that to suppose that there is ever any discontinuity in change. That is to say, between any two instantaneous states there must be a lapse of time during which the change is continuous, not merely in that false continuity which the calculus recognizes but in a much stricter sense. — Peirce, R 300, 1908

Sounds like Peirce is rejecting discontinuous functions. Or something like that. I know that in constructive math, all functions are computably continuous or something like that. Makes some of the problems go away.

That particular text is incomplete and largely unpublished, but there is a transcription online — aletheist

Oh jeez I gave that link a look. Not exactly a clear writer IMO. I don't think I could ever slog through this.

You might be interested in John Bell's book, The Continuous and the Infinitesimal in Mathematics and Philosophy. — aletheist

Dispatching a clone. Not the first time I've had this book recommended to me. Sigh.

And the defining characteristic involving squared infinitesimals seems just another strange notion one can avoid — jgill

I always thought this was an abstraction of basic 17th century calculus, where higher powers of infinitesimals can be ignored.

There is nothing inherently contradictory about the mathematical concept of an infinitesimal, which is not necessarily defined as 1/∞. Again, if you truly want to understand, please read one or both of the short articles that I linked. If you prefer to remain ignorant, carry on. — aletheist

The problem with the articles you linked is that in both cases, a wrong assumption is made at the start of discourse:

"In order for SDG to be consistent, the law of exclude middle must not hold. SDG does not rely on classical logic but on intuitionistic logic."

I happen to strongly believe that the LEM holds for our universe and indeed all possible universes.

"Now if it were possible to take δx so small (but not demonstrably identical with 0 that (δx)^2 = 0"

I also strongly believe there is no nonzero x such that x^2=0.

Why should I invest time and effort learning subjects that are based on wrong assumptions? All 'knowledge' I'd acquire in doing so would be inherently unsound.

You physically can't keep putting 10 balls in a vase, while only removing one. It's an unrealistic thought experiment.
— Harry Hindu

Why is it unrealistic? — Devans99

I explained why. Pay attention. What vase would be large enough to keep putting in 10 balls while only removing one? The vase would have to be an infinite sized container, which makes no sense. How can something be both infinite and contain?

I don't believe everything came from nothing, I believe that something has permanent, atemporal existence and that something caused everything else — Devans99

If it has an atemporal existence then that is the same as saying that it doesn't exist. How does something cause everything else without being in time itself? How does it cause anything without changing itself? Even God has to exist in time if God changes. Change is time.

Double post

The problem with the articles you linked is that in both cases, a wrong assumption is made at the start of discourse: — Devans99

There are no "wrong" assumptions in pure mathematics. It is the science of reasoning necessarily about hypothetical states of things.

I happen to strongly believe that the LEM holds for our universe and indeed all possible universes. — Devans99

LEM holds for anything that is determinate, including anything that is discrete; but it does not hold for anything that is indeterminate, including anything that is truly continuous. There is nothing illegitimate about intuitionistic logic.

I also strongly believe there is no nonzero x such that x^2=0. — Devans99

What you believe is irrelevant. There is nothing self-contradictory about defining an infinitesimal as that which is not itself equal to zero, but whose squares and higher powers are equal to zero; and it turns out to be quite useful.

Why should I invest time and effort learning subjects that are based on wrong assumptions? All 'knowledge' I'd acquire in doing so would be inherently unsound. — Devans99

The problem is treating assumptions other than your own as indubitably wrong and refusing even to entertain them, which is a textbook example of sheer dogmatism. Unless you consider yourself to be infallible, you might want to try opening your mind a bit.

Even God has to exist in time if God changes. Change is time. — Harry Hindu

Good point. Time is eternity; eternity time.

I suppose that's one of the vexing problems in making sense of an unchanging Being or thing, in a world of change. Time seems to be an abstract reality, yet no different than other abstract realities that exist and are perceived in consciousness [conscious existence] like the existence of mathematics itself, etc..

I know what you mean about clones! Hopefully you can at least digest the two short articles about SDG and SIA. Browsing Moore's single volume of Peirce's writings on philosophy of mathematics might be enough to give you the gist of his overall approach and help you decide whether delving deeper is worth the trouble.

There are no "wrong" assumptions in pure mathematics. It is the science of reasoning necessarily about hypothetical states of things. — aletheist

Fair point, but assumptions that stray wildly from common sense / common experience indicate the subject is squarely pure rather than applied maths. IE it tells us nothing about our reality, it is telling us something about an alternative reality where common sense does not apply. I am not interested in maths for maths sake, I am interested in what it tells us about the reality we live in. If parts of maths adopt axioms that depart from common sense, then I have to disregard those parts when searching for a description of our reality.

This is the way you should approach maths (and other fields of human knowledge) - you look at the axioms and decide if you believe them or not. Then you learn about the parts that you believe have sound axioms and disregard the rest. Parts of math claim that for non-zero x, x^2>0. Other parts of maths claim that for non-zero x, x^2=0. The two parts of math are incompatible. It is only possible to hold a belief in one of these two incompatible parts of math. I put my money on basic arithmetic.

If parts of maths adopt axioms that depart from common sense, then I have to disregard those parts when searching for a description of our reality. — Devans99

Common sense tells us that common sense is highly fallible. Some developments in mathematics and science over the centuries are highly counterintuitive, and if we had insisted on sticking with common sense, we would still be misunderstanding reality. Is it common sense that there are numbers incapable of being calculated as fractions of integers? Or that matter consists of atoms that in turn consist of varying quantities of protons, neutrons, and electrons? Or that gravity is the curvature of spacetime, rather than a direct force of attraction between massive bodies?

It is only possible to hold a belief in one of these two incompatible parts of math. — Devans99

Nonsense, there is no single set of mathematical assumptions that perfectly matches reality--just different models that are useful for different purposes.

I put my money on basic arithmetic. — Devans99

Do parallel lines ever meet?

I explained why. Pay attention. What vase would be large enough to keep putting in 10 balls while only removing one. The vase works have to be an infinite sized container, which makes no sense. How can something be both infinite and contain? — Harry Hindu

The paradox starts with the assumption that actual infinity is possible, so it is OK to assume an actually infinite bag/vase.

If has an atemporal existence then that is the same as saying that it doesn't exist. How foes something cause everything else without being in time itself? How does it cause anything without changing itself? Even God has to exist in time if God changes. Change is time. — Harry Hindu

I have done a probability analysis of all the arguments and I get 94% certain that time has a start.
That implies something atemporal must very probably exist in order to be the cause of time. I imagine the whole of the universe as a 2d spacetime diagram of finite size. Then I imagine the atemporal thing (God) off to the side (not on the plane) and a mapping between the atemporal thing and each point in the plane. Then the atemporal thing can express itself in spacetime without being part of spacetime.

I am not sure precisely how the atemporal thing (God) could work. But then if you think about all the universes in the multiverse, all the multiverses in reality and all of the different possible realities that might exist, it seems impossible that we would ever understand them all - so things with a drastically different nature very probably exist - including atemporal things.

Time enables change. Time is not change. If time was change then time would flow faster in the presence of change, yet SR indicates time slows down in the presence of change.

Do parallel lines ever meet? — tim wood

Who knows. I don't disregard non-euclidean geometry because its axiom that parallel lines meet has a possibility of being true. I do however disregard maths that does not follow the LEM. There is so much maths, all different fields with different axioms that disagree with each other - it is therefore required to be selective in what one chooses when trying to use maths to understand the universe.

Is it common sense that there are numbers incapable of being calculated as fractions of integers? Or that matter consists of atoms that in turn consist of varying quantities of protons, neutrons, and electrons? Or that gravity is the curvature of spacetime, rather than a direct force of attraction between massive bodies? — aletheist

The proof of irrational numbers is common sense. We have empirical evidence for atoms and the curvature of spacetime. So these things are in agreement with common sense.

Nonsense, there is no single set of mathematical assumptions that perfectly matches reality--just different models that are useful for different purposes. — aletheist

But as seekers of a truthful explanation of our reality, we have to make choices between incompatible branches of mathematics. I'm unwilling to discard arithmetic from my set of choices of valid mathematics.

The proof of irrational numbers is common sense. We have empirical evidence for atoms and the curvature of spacetime. So these things are in agreement with common sense. — Devans99

We clearly have very different definitions of "common sense."

But as seekers of a truthful explanation of our reality, we have to make choices between incompatible branches of mathematics. I'm unwilling to discard arithmetic from my set of choices of valid mathematics. — Devans99

Who said anything about discarding arithmetic? It is very useful for very many purposes, especially those encountered in everyday life, which generally involve dealing with finite quantities of discrete things. A different approach is required to handle potentially infinite sets, and yet another is required for true continuity. Whether this accords with "common sense" or not, it is the reality.

Who said anything about discarding arithmetic? It is very useful for very many purposes, especially those encountered in everyday life, which generally involve dealing with finite quantities of discrete things. A different approach is required to handle potentially infinite sets, and yet another is required for true continuity. Whether this accords with "common sense" or not, it is the reality. — aletheist

Your chosen version of reality included continua (incompatible with arithmetic IMO) and infinitesimals (incompatible with arithmetic IMO). My chosen version of reality does not include these two concepts (I think reality is finite and discrete). Time will tell which version of reality is correct.

No one gets to choose a "version of reality," because by definition reality is as it is regardless of what anyone thinks about it. Adopting finite discrete arithmetic as the sole or primary basis for your entire metaphysics strikes me as extremely naive. I hope that someday you will be more open to alternatives.

That implies something atemporal must very probably exist in order to be the cause of time. — Devans99

Unchanging causing change is as incoherent as something coming from nothing.

Then I imagine the atemporal thing (God) off to the side (not on the plane) and a mapping between the atemporal thing and each point in the plane. Then the atemporal thing can express itself in spacetime without being part of spacetime. — Devans99

See? You can't escape talking about God relative to the universe. You are implying space-time encompassing God and your universe, as God is located relative to the universe and expresses itself in time.

But then if you think about all the universes in the multiverse, all the multiverses in reality and all of the different possible realities that might exist, it seems impossible that we would ever understand them all - so things with a drastically different nature very probably exist - including atemporal things. — Devans99

They exist only as imaginings in the human mind in this particular universe.

Time enables change. Time is not change. If time was change then time would flow faster in the presence of change, yet SR indicates time slows down in the presence of change. — Devans99

I dont know what you're talking about. Maybe you're talking about realtive change. There is more or less change in one area relative to another.

Unchanging causing change is as incoherent as something coming from nothing. — Harry Hindu

I did not say timeless beings cannot cause change. I said that a timeless being can express itself in spacetime and thereby be the agent of change.

See? You can't escape talking about God relative to the universe. You are implying space-time encompassing God and your universe, as God is located relative to the universe and expresses itself in time. — Harry Hindu

I am not implying space-time encompassing God, I said God is external to spacetime but can express himself in spacetime.

Spacetime is fine-tuned for life. That requires a fine-tuner from beyond spacetime. Nothing can exist forever in time, that requires a first cause from beyond time.

They exist only as imaginings in the human mind in this particular universe. — Harry Hindu

You cannot possibly have understanding of every possible reality beyond our own, so you cannot make such a claim.

I dont know what you're talking about. Maybe you're talking about realtive change. There is more or less change in one area relative to another. — Harry Hindu

What I mean is that special relativity says that time is observed to slow down as things move faster (hence the photon moving at the speed of light is a timeless particle). So more change = less time. So time is not change (because that would lead to more change = more time - in opposition to special relativity).

I always thought this was an abstraction of basic 17th century calculus, where higher powers of infinitesimals can be ignored. — fishfry

True enough. In computations involving non-infinitesimal calculus higher order terms can be ignored depending on the settings.

I know that in constructive math, all functions are computably continuous or something like that. Makes some of the problems go away. — fishfry

Hmmm. I keep learning things here. Thanks. :chin:

https://people.eecs.berkeley.edu/~fateman/papers/limit.pdf

My argument uses sequences of identical bananas, so that the 'quantity' and 'quality' of bananas both are constant whilst bananas are added and removed from the sequences - resulting in absurdity. — Devans99

Well then you're contradicting yourself. Things can change either qualitatively or quantitatively and you say neither has occurred. Then in what way have the sets changed; after all your claim is that when it is changed, it is not changed.

Well then you're contradicting yourself. Things can change either qualitatively or quantitatively and you say neither has occurred. Then in what way have the sets changed; after all your claim is that when it is changed, it is not changed. — TheMadFool

The argument in the OP is that you can add/remove identical items to an infinite sequence and the sequence remains identical/unchanged (both qualitatively and quantitatively). This results in the contradiction 'when it is changed, it is not changed', which is what I intended - assumption of the existence of actual infinity leads to a contradiction.

The argument in the OP is that you can add/remove identical items to an infinite sequence and the sequence remains identical/unchanged (both qualitatively and quantitatively). This results in the contradiction 'when it is changed, it is not changed', which is what I intended - assumption of the existence of actual infinity leads to a contradiction. — Devans99

So add/remove is the change. How? In what way have you changed the infinite set from which something has been removed and the infinite set to which something has been added? You deny that the change is qualitative since you claim you're using identical bananas. You deny that it is a quantitative change since infinity + 1 = infinity and infinity/2 = infinity. So, it must be that nothing has changed and that precludes any claim that the sets have changed in any way.

Is change that is neither qualitative nor quantitative possible?

Also I have an issue of identicalness of the bananas. How does this identicalness weigh in on the issue? Since you've listed your sets as {b, b, b,...} it implies that there's a difference between any two b's; after all if they were logically identical in that all b's refer to one and only one object then you wouldn't or rather couldn't list them separately as {b, b, b,...}; set theory doesn't allow repetitions of elements. Since the difference between b's isn't quantitative because b corresponds to the number 1 it follows that the b's differ qualitatively and that points to a qualitative change when you manipulate the two sets as you do.

If objects are infinitely divisible, then an actual infinity is real. I don't see how finite geometry can be defended. Euclid and Archimedes understood this

So add/remove is the change. How? In what way have you changed the infinite set from which something has been removed and the infinite set to which something has been added? — TheMadFool

We add 1 banana to the sequence (=it should change quantitatively and qualitatively).
But is does not change quantitatively(∞+1=∞) or qualitatively(still identical rows of identical bananas).
So there is something that when we change it, it does not change
That's a contradiction so our premise must be wrong
Hence actual infinity is impossible.

set theory doesn't allow repetitions of elements — TheMadFool

Thats why they are sequences rather than sets (sequences allow duplicate objects).