Infinitesimals are deeply illogical/impossible concepts and are shunned by most of maths. — Devans99
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
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
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
either absolutely unchanging or always changing continuously. — Peirce, R 300, 1908
For another, functions in SDG/SIA "are differentiable arbitrarily many times" — aletheist
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.So just as I call modern constructivism Brouwers revenge, I can call SIA Peirce's revenge. — 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.Do you happen to know how SIA relates to constructive math? They both deny excluded middle as I understand it. — 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 That particular text is incomplete and largely unpublished, but there is a transcription online.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.What's R 300? Where can I find Peirce talking about calculus? — fishfry
Here is what Peirce wrote right before the sentence that I quoted.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
Presumably the "false continuity" that he had in mind was that of Cantor. Does this help at all?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
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.Why are there so many Peirceans on this forum? I never hear about him anywhere else but his ideas are incredibly interesting. — fishfry
so I would welcome your thoughts on SDG/SIA — aletheist
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
That particular text is incomplete and largely unpublished, but there is a transcription online — aletheist
You might be interested in John Bell's book, The Continuous and the Infinitesimal in Mathematics and Philosophy. — aletheist
And the defining characteristic involving squared infinitesimals seems just another strange notion one can avoid — jgill
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
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?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
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.I don't believe everything came from nothing, I believe that something has permanent, atemporal existence and that something caused everything else — Devans99
There are no "wrong" assumptions in pure mathematics. It is the science of reasoning necessarily about hypothetical states of things.The problem with the articles you linked is that in both cases, a wrong assumption is made at the start of discourse: — 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 happen to strongly believe that the LEM holds for our universe and indeed all possible universes. — 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.I also strongly believe there is no nonzero x such that x^2=0. — 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.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
Even God has to exist in time if God changes. Change is time. — Harry Hindu
There are no "wrong" assumptions in pure mathematics. It is the science of reasoning necessarily about hypothetical states of things. — aletheist
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?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
Nonsense, there is no single set of mathematical assumptions that perfectly matches reality--just different models that are useful for different purposes.It is only possible to hold a belief in one of these two incompatible parts of math. — Devans99
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
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
Do parallel lines ever meet? — tim wood
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
Nonsense, there is no single set of mathematical assumptions that perfectly matches reality--just different models that are useful for different purposes. — aletheist
We clearly have very different definitions of "common sense."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
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.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. — aletheist
Unchanging causing change is as incoherent as something coming from nothing.That implies something atemporal must very probably exist in order to be the cause of time. — 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.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
They exist only as imaginings in the human mind in this particular universe.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
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.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
Unchanging causing change is as incoherent as something coming from nothing. — Harry Hindu
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
They exist only as imaginings in the human mind in this particular universe. — Harry Hindu
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
I always thought this was an abstraction of basic 17th century calculus, where higher powers of infinitesimals can be ignored. — fishfry
I know that in constructive math, all functions are computably continuous or something like that. Makes some of the problems go away. — fishfry
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. — 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. — 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? — TheMadFool
set theory doesn't allow repetitions of elements — TheMadFool
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