Isn't the definition of infinitesimal contradictory?

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bahman
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Re: Isn't the definition of infinitesimal contradictory?

Post by bahman »

wtf wrote: Wed May 23, 2018 7:00 pm
bahman wrote: Wed May 23, 2018 6:44 pm I am afraid that I don't have access to an article from him on this subject. The wiki source was the best I could find quickly.
But the WIkipedia page doesn't say that Leibniz said what you say he did.

May I take it that you retract your claim?
bahman wrote: Wed May 23, 2018 6:44 pm And yes. I would like to discuss about infinitesimal.
Ok. The mathematical definition of an infinitesimal is a positive number that is less than 1/n for any positive integer n.

There is obviously no such thing in the real numbers. No matter how small some claimed infinitesimal is, we can look at 1/10, 1/100, 1.1000, etc., and eventually find an n such that 1/n is smaller than the claimed infinitesimal.

By assuming a weak form of the axiom of choice, one can show the existence of a gadget called a nonprincipal ultrafilter, which one can then use to produce a field (a mathematical system in which we can add, subtract, multiply, and divide) in which there are infinitesimals. This field is called the hyperreals. [Technically there are many such fields since different nonprincipal ultrafilters give rise to nonisomorphic fields of hyperreals. And it's a mathematical curiosity that the Continuum Hypothesis implies that there is only one unique field of hyperreals. As you can see we are in deep foundational waters].

There's also a construction call the surreals that contains infinitesimals.

As I noted above, neither the hyperreals nor the surreals are topologically complete; and the surreals are not even a set. And as noted, the hyperreals require stronger set theoretic axioms than do the standard reals.

So the standard reals are standard for good reason. And even pedagogically, studies don't show any advantage for teaching freshman calculus via hyperreals versus the standard reals. The students come out confused either way. This is why the standard reals persist in mathematics and in mathematical pedagogy. Systems with infinitesimals don't give any benefit. It's fair to note that the hyperreal zealots claim students prefer infinitesimal-based calculus. My reading shows that studies are inconclusive at best.

Now Leibniz did make some interesting philosophical points about the nature of infinitesimals, but I'm not qualified to discuss them since I haven't studied Leibniz. I have studied the hyperreals a bit so I can answer questions about them if you have any.

I would also add that the Wiki article on infinitesimals is not very good and is frequently quote-mined to ill effect in online discussions.
Are you saying that we still don't know infinitesimal (considering bold part)?
wtf
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Re: Isn't the definition of infinitesimal contradictory?

Post by wtf »

bahman wrote: Thu May 24, 2018 5:36 pm Are you saying that we still don't know infinitesimal (considering bold part)?
I was talking about the effects of infinitesimals on teaching freshman calculus. There have been some efforts to teach calculus using infinitesimals instead of the standard epsilon-delta approach. Some proponents of the infinitesimal approach say students like it better. The studies I've seen don't support that. Students come out confused either way.

The remark you questioned was about teaching, not about the nature of mathematical infinitesimals.
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Re: Isn't the definition of infinitesimal contradictory?

Post by wtf »

Eodnhoj7 wrote: Thu May 24, 2018 3:50 pm And what differs this from what Leibniz "might have said"?
Leibniz did not write the Wikipedia article, which in fact is riddled with errors and confusing exposition.
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Re: Isn't the definition of infinitesimal contradictory?

Post by Eodnhoj7 »

wtf wrote: Thu May 24, 2018 5:43 pm
Eodnhoj7 wrote: Thu May 24, 2018 3:50 pm And what differs this from what Leibniz "might have said"?
Leibniz did not write the Wikipedia article, which in fact is riddled with errors and confusing exposition.
Okay, what differs in your opinion?
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Re: Isn't the definition of infinitesimal contradictory?

Post by wtf »

Eodnhoj7 wrote: Thu May 24, 2018 5:44 pm Okay, what differs in your opinion?
Everything. Leibniz never said any of the things the Wiki author said.
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Re: Isn't the definition of infinitesimal contradictory?

Post by Eodnhoj7 »

wtf wrote: Thu May 24, 2018 5:52 pm
Eodnhoj7 wrote: Thu May 24, 2018 5:44 pm Okay, what differs in your opinion?
Everything. Leibniz never said any of the things the Wiki author said.
I looked at some of his notes and they, while not exact as you said, mirror heavily this relativistic fractal approach...I would argue they are a correct "interpretation" (emphasis on interpretation).

His notes also had a lot of stuff marked out in them...

What would you argue he specifically said? Maybe I need different sources too.
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Re: Isn't the definition of infinitesimal contradictory?

Post by wtf »

Eodnhoj7 wrote: Fri May 25, 2018 12:09 am I looked at some of his notes and they, while not exact as you said, mirror heavily this relativistic fractal approach...I would argue they are a correct "interpretation" (emphasis on interpretation).

His notes also had a lot of stuff marked out in them...

What would you argue he specifically said? Maybe I need different sources too.
Perhaps you could link or at least identify your sources so others could see for themselves. Surely old Gottfried didn't edit Wikipedia. I have no idea why the OP copy/pasted a block of Wiki text and attributed it to Leibniz and I don't understand why you are making the same claim. A glance at the Wiki page shows this to be flat out false.
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Re: Isn't the definition of infinitesimal contradictory?

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wtf wrote: Fri May 25, 2018 12:52 am I have no idea why the OP copy/pasted a block of Wiki text and attributed it to Leibniz and I don't understand why you are making the same claim. A glance at the Wiki page shows this to be flat out false.
The reason is extremely simple. The religious nuts are incapable of human thought. They are incapable of normal discourse. They are incapable of deciphering the language. They have no knowledge of what it means to cite an original statement.

I have come to the conclusion which I stay firmly by, that the religious nuts on this forum are not worth discussing anything with.

I started to boycott all debates for this very reason itself.
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Re: Isn't the definition of infinitesimal contradictory?

Post by wtf »

-1- wrote: Fri May 25, 2018 2:14 am The reason is extremely simple. The religious nuts are incapable of human thought.
I don't see it that way at all. Infinitesimals are a deep and unfathomable concept that has puzzled thinkers since antiquity. In the 19th century mathematicians finessed the issue with the theory of limits. But the ancient idea of infinitesimals hasn't gone away; and in modern times (post-1948) infinitesimals have finally been put on a sound mathematical footing.

So it's perfectly reasonable for people to be curious about infinitesimals.

Now as it happens, the Wikipedia article on the subject is atrocious, filled with contradictions, misinformation, and confusion. It's fairly normal for someone to read that page and come to an online forum with questions. The very first sentence, defining an infinitesimal as something that can't be measured, is garbage. The word measure has a very specific meaning in math. There are non-measurable sets, and they are most definitely not infinitesimals. And putting it this way confuses math with physics, since in physics we do have a limit below which things can't be measured. And even though it's true in everyday speech that we call something infinitesimal when it's really really small, is irrelevant. It's not math. It's just colloquial English and only muddies the waters to bring it up at all in a mathematical context.

In addition, Leibniz did talk about infinitesimals and his related philosophical concept, monads. My knowledge of the history of calculus is much stronger regarding Newton than Leibniz. So I hope to help to shed some light on the subject and perhaps learn something about Leibniz's thought.
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Re: Isn't the definition of infinitesimal contradictory?

Post by Eodnhoj7 »

-1- wrote: Fri May 25, 2018 2:14 am
wtf wrote: Fri May 25, 2018 12:52 am I have no idea why the OP copy/pasted a block of Wiki text and attributed it to Leibniz and I don't understand why you are making the same claim. A glance at the Wiki page shows this to be flat out false.
The reason is extremely simple. The religious nuts are incapable of human thought. They are incapable of normal discourse. They are incapable of deciphering the language. They have no knowledge of what it means to cite an original statement.

I have come to the conclusion which I stay firmly by, that the religious nuts on this forum are not worth discussing anything with.

I started to boycott all debates for this very reason itself.
Last time I observed secularism argued for the embracement of all dispositions...and where is religion brought up at all in this? Take your self-righteous hatred elsewhere...unless you can provide an argument for it. Even then this is the wrong thread.

The only person who cares about your boycott is you considering the arguments you provide...well...you did your best I guess that is what counts.
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Re: Isn't the definition of infinitesimal contradictory?

Post by Eodnhoj7 »

wtf wrote: Fri May 25, 2018 12:52 am
Eodnhoj7 wrote: Fri May 25, 2018 12:09 am I looked at some of his notes and they, while not exact as you said, mirror heavily this relativistic fractal approach...I would argue they are a correct "interpretation" (emphasis on interpretation).

His notes also had a lot of stuff marked out in them...

What would you argue he specifically said? Maybe I need different sources too.
Perhaps you could link or at least identify your sources so others could see for themselves. Surely old Gottfried didn't edit Wikipedia. I have no idea why the OP copy/pasted a block of Wiki text and attributed it to Leibniz and I don't understand why you are making the same claim. A glance at the Wiki page shows this to be flat out false.
Second post of mine, page 2:
I looked at some of his notes and they, while not exact as you said, mirror heavily this relativistic fractal approach...I would argue they are a correct "interpretation" (emphasis on interpretation).

http://www.leibniz-translations.com/pascal.htm

The actual infinity in material things, as much as in the increasingly large as in the vanishingly small,1 that is, the actual division of each part of matter to infinity and at the same time the infinite vastness of matter, has been supported by Mr Pascal, and it is evident that those who have contemplated his Pensées, as well as the bishops and doctors who have approved it, have agreed with it. Here is one of the passages which show it: it is at number 22, entitled General knowledge of man:
'The first thing which presents itself to man when he looks at himself is his body ... to abysses.'2
This is as far as Mr Pascal goes.3
What Mr. Pascal says about the double infinity, which surrounds us in the increasingly large and the vanishingly small, when in his Pensées [Gr p554] (n.22) he talks about the general knowledge of man, is only an entrance into my system. What would he not have said with that power of eloquence he possessed if he had gone further, if he had known that all matter is organic everywhere, and that, however small a portion one takes, it contains representatively, by virtue of the actual decreasing to infinity that it encloses, the actual increasing to infinity which is outside it in the universe. That is, each small portion contains, in an infinity of ways, a living mirror expressing the whole infinite universe that exists with it; so that a sufficiently great mind, armed with a sufficiently penetrating view, could see here everything everywhere. But there is much more: it could even read the whole of the past there, and even the whole infinitely infinite future, since each moment contains an infinity of things, each of which envelops an infinity, and since there is an infinity of moments in each hour or other part of time, and an infinity of hours, of years, of centuries and eons in the whole of future eternity. What an infinity of infinities infinitely replicated, what a world, what a universe perceptible in any assignable corpuscle. But all these wonders are surpassed by the envelopment of what is infinitely above all greatnesses in what is infinitely below all smallnesses. That is, our pre-established harmony, which has only recently appeared on the scene, and which yields even more than absolutely universal infinity, concentrated in the more than infinitely small and absolutely singular, by placing, virtually, the whole series of the universe in each real point which makes a Monad or substantial unity, of which I am one. That is, in each substance truly one, unique, primitive subject of life and action, always endowed with perception and with appetition, always containing in what it is the tendency to what it will be,4 to represent everything else which will be.5 The basic almost-nothing, in coming up from nothing to things, since it is the simplest of them, is also as it were the highest almost-everything [Gr p555] in descending from the multitude of things towards nothing; and yet it is the only thing that deserves to be called a being, a substance after God, because a multitude is only a mass of several substances, and not a being, but beings. It is this simple and primitive subject of tendencies and actions, this interior source of its own changes, which is therefore the only way of true imperishable being, since it is indissoluble and without parts, always subsisting and which will never perish, any more than will God and the universe, which it must always represent and in whole.6 And if this Monad is a mind, that is, a soul capable of reflection and knowledge,7 it will be at the same time infinitely less than a God and incomparably more than the rest of the universe of creatures; it senses everything confusedly, unlike God who knows everything distinctly; it knows something distinctly, unlike the whole of matter which knows and senses nothing of the whole. It will be a diminutive divinity and a universe of matter eminently; God in ectype and this universe in prototype, as the intelligible is always anterior to the sensible in the ideas of the primitive intelligence, the source of things; it will imitate God and be imitated by the universe in relation to its distinct thoughts. It will be subject to God in everything, and dominator of all creatures to the extent that it is an imitator of God.




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wtf
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Re: Isn't the definition of infinitesimal contradictory?

Post by wtf »

Eodnhoj7 wrote: Fri May 25, 2018 3:23 pm
Do you want more?
I'd like less, but on point.
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bahman
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Re: Isn't the definition of infinitesimal contradictory?

Post by bahman »

Impenitent wrote: Wed May 23, 2018 9:44 pm half a monad doesn't count

-Imp
Isn't a monad an infinitesimal? What is your definition monad?
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Re: Isn't the definition of infinitesimal contradictory?

Post by bahman »

wtf wrote: Fri May 25, 2018 12:52 am I have no idea why the OP copy/pasted a block of Wiki text and attributed it to Leibniz and I don't understand why you are making the same claim. A glance at the Wiki page shows this to be flat out false.
What is your definition of infinitesimal?
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bahman
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Re: Isn't the definition of infinitesimal contradictory?

Post by bahman »

-1- wrote: Fri May 25, 2018 2:14 am
wtf wrote: Fri May 25, 2018 12:52 am I have no idea why the OP copy/pasted a block of Wiki text and attributed it to Leibniz and I don't understand why you are making the same claim. A glance at the Wiki page shows this to be flat out false.
The reason is extremely simple. The religious nuts are incapable of human thought. They are incapable of normal discourse. They are incapable of deciphering the language. They have no knowledge of what it means to cite an original statement.

I have come to the conclusion which I stay firmly by, that the religious nuts on this forum are not worth discussing anything with.

I started to boycott all debates for this very reason itself.
Could you please elaborate further and define infinitesimal?
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