Isn't the definition of infinitesimal contradictory?

[bahman]

According to Leibniz, infinitesimal is the smallest non-zero number. How such a thing can exist? I think for any real number, x, there exists a real number, y, where y<x.

[Eodnhoj7]

According to Leibniz, infinitesimal is the smallest non-zero number. How such a thing can exist? I think for any real number, x, there exists a real number, y, where y<x.

Leibniz is highly relativistic if you look at any of his work and some historians believed he was the preemptive discoverer of relativity that later inspired einstein.

Because Leibniz is a relativist, I would argue (this is me speaking) that the smallest non-zero number would have to be observed as

(1>n → 0) = (1/1 → 1/2 → 1/3 → ∞) = 1/(1 → 2 → 3 → ∞) as a continual fraction. In this manner the "number" would equate to more of a fractal series. These fractions, a as a continual form of self division require whole numbers (through 1 folding through itself as 2,3, etc. as its own ratios).

This inversion of the fractals of one leads simultaneously not just to whole numbers but a continual series approaching away from zero.

The problem of a series is that is follows the form of a number line, unless someone understands it differently, and while the line may extend from point zero it simultaneously extends back to it.

So the fractals which manifest as a result of moving towards point zero, when inverted, result in whole numbers being formed "because of" a move towards 0.

The problem occurs in that this series of numbers in itself is continually approaching "smallness" as "smallness" is merely a term of relation in regards to size. I would argue, unless the number is observed specifically, that what he is observing (intentionally or not) is a process of fractation and not a fraction itself specifically.

Relativism, as the relation of parts expressed at best through fractals, is dependent upon a series of continual movement.

[wtf]

According to Leibniz, infinitesimal is the smallest non-zero number.

Leibniz surely never said any such thing.

[bahman]

According to Leibniz, infinitesimal is the smallest non-zero number.

Leibniz surely never said any such thing.

He of course said that (please notice the bold part) :" In mathematics, infinitesimals are things so small that there is no way to measure them. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were quantitatively small.[1] The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a sequence. Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" means "extremely small". To give it a meaning, it usually must be compared to another infinitesimal object in the same context (as in a derivative). Infinitely many infinitesimals are summed to produce an integral."

[bahman]

According to Leibniz, infinitesimal is the smallest non-zero number. How such a thing can exist? I think for any real number, x, there exists a real number, y, where y<x.

Leibniz is highly relativistic if you look at any of his work and some historians believed he was the preemptive discoverer of relativity that later inspired einstein.

Because Leibniz is a relativist, I would argue (this is me speaking) that the smallest non-zero number would have to be observed as

(1>n → 0) = (1/1 → 1/2 → 1/3 → ∞) = 1/(1 → 2 → 3 → ∞) as a continual fraction. In this manner the "number" would equate to more of a fractal series. These fractions, a as a continual form of self division require whole numbers (through 1 folding through itself as 2,3, etc. as its own ratios).

This inversion of the fractals of one leads simultaneously not just to whole numbers but a continual series approaching away from zero.

The problem of a series is that is follows the form of a number line, unless someone understands it differently, and while the line may extend from point zero it simultaneously extends back to it.

So the fractals which manifest as a result of moving towards point zero, when inverted, result in whole numbers being formed "because of" a move towards 0.

The problem occurs in that this series of numbers in itself is continually approaching "smallness" as "smallness" is merely a term of relation in regards to size. I would argue, unless the number is observed specifically, that what he is observing (intentionally or not) is a process of fractation and not a fraction itself specifically.

Relativism, as the relation of parts expressed at best through fractals, is dependent upon a series of continual movement.

I also think that there is a relation between infinitesimal and fractal as you describe. Is this series, 1/x^2 also a fractal in the limit when x->infinity?

[wtf]

According to Leibniz, infinitesimal is the smallest non-zero number.

Leibniz surely never said any such thing.

He of course said that (please notice the bold part) :" In mathematics, infinitesimals are things so small that there is no way to measure them. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were quantitatively small.[1] The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a sequence. Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" means "extremely small". To give it a meaning, it usually must be compared to another infinitesimal object in the same context (as in a derivative). Infinitely many infinitesimals are summed to produce an integral."

Your post doesn't support your claim.

[bahman]

Leibniz surely never said any such thing.

He of course said that (please notice the bold part) :" In mathematics, infinitesimals are things so small that there is no way to measure them. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were quantitatively small.[1] The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a sequence. Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" means "extremely small". To give it a meaning, it usually must be compared to another infinitesimal object in the same context (as in a derivative). Infinitely many infinitesimals are summed to produce an integral."

Your post doesn't support your claim.

Isn't smallest feasible measurement is related to smallest number?

[wtf]

Isn't smallest feasible measurement is related to smallest number?

You've already changed the subject.

You did not make a claim about mathematics. You made a claim about history. You said that "Leibniz said X." It doesn't matter what anyone else says about X, or what the merits of X itself are, nor does quote-mining Wiki for what OTHER people said about X help at all.

Leibniz did not say what you claim he did. Even the Wiki article on infinitesimals fails to support your assertion.

There are two things going on here.

One is the nature of infinitesimals. We could talk about that if you like. Here's the long and short of it. There are no infinitesimals in the real numbers. There are systems such as the hyperreals and the surreals in which there are infinitesimals. All such systems suffer from the flaw that they are not topologically complete. The least upper bound property is false in any number field containing infinitesimals. In addition, the hyperreals require stronger set theoretical axioms than the standard reals do; and the surreals are not a set. So the standard reals have a lot of mathematical and philosophical power that is not diminished by the existence of systems containing infinitesimals.

But you made a claim that Leibniz said something that he did not and could not have said.

Leibniz did not and could not have said that an infinitesimal is the smallest positive real number, for the obvious reason that dividing any such claimed infinitesimal by 2 would result in a smaller positive real number. Leibniz would have known that, and evidently you yourself know that since you pointed it out in your OP.

Leibniz did not say what you claim he said. By falsely attributing a manifestly false statement to Leibniz and then arguing against that statement, you are engaging in a strawman argument.

So why don't you either support with evidence your claim that Leibniz said what you claim he said; or else retract it. If you want to talk about infinitesimals let's do that. But there's no point in saying, "Von Neuman said 2 + 2 = 5, wasn't he a dope?" when in fact von Neumann never said any such thing.

[bahman]

Isn't smallest feasible measurement is related to smallest number?

You've already changed the subject.

You did not make a claim about mathematics. You made a claim about history. You said that "Leibniz said X." It doesn't matter what anyone else says about X, or what the merits of X itself are, nor does quote-mining Wiki for what OTHER people said about X help at all.

Leibniz did not say what you claim he did. Even the Wiki article on infinitesimals fails to support your assertion.

There are two things going on here.

One is the nature of infinitesimals. We could talk about that if you like.

But first you need to either support with evidence or else retract your claim that Leibniz said X, when in fact your own post shows that X in this case is manifestly false and that Leibniz, a good mathematician, would have known that.

Leibniz did not and could not have said that an infinitesimal is the smallest positive real number, for the obvious reason that dividing any such claimed infinitesimal by 2 would result in a smaller positive real number. Leibniz would have known that, and evidently you yourself know that.

Leibniz did not say what you claim he said. By falsely attributing a manifestly false statement to Leibniz and then arguing against that statement, you are engaging in a strawman argument.

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.

And yes. I would like to discuss about infinitesimal.

[wtf]

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?

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.

[Impenitent]

half a monad doesn't count

-Imp

[wtf]

Isn't smallest feasible measurement is related to smallest number?

No, that confusing math with physics.

There's no smallest positive real number. But in modern physics there's a smallest possible measurement.

[Eodnhoj7]

According to Leibniz, infinitesimal is the smallest non-zero number. How such a thing can exist? I think for any real number, x, there exists a real number, y, where y<x.

Leibniz is highly relativistic if you look at any of his work and some historians believed he was the preemptive discoverer of relativity that later inspired einstein.

Because Leibniz is a relativist, I would argue (this is me speaking) that the smallest non-zero number would have to be observed as

(1>n → 0) = (1/1 → 1/2 → 1/3 → ∞) = 1/(1 → 2 → 3 → ∞) as a continual fraction. In this manner the "number" would equate to more of a fractal series. These fractions, a as a continual form of self division require whole numbers (through 1 folding through itself as 2,3, etc. as its own ratios).

This inversion of the fractals of one leads simultaneously not just to whole numbers but a continual series approaching away from zero.

The problem of a series is that is follows the form of a number line, unless someone understands it differently, and while the line may extend from point zero it simultaneously extends back to it.

So the fractals which manifest as a result of moving towards point zero, when inverted, result in whole numbers being formed "because of" a move towards 0.

The problem occurs in that this series of numbers in itself is continually approaching "smallness" as "smallness" is merely a term of relation in regards to size. I would argue, unless the number is observed specifically, that what he is observing (intentionally or not) is a process of fractation and not a fraction itself specifically.

Relativism, as the relation of parts expressed at best through fractals, is dependent upon a series of continual movement.

I also think that there is a relation between infinitesimal and fractal as you describe. Is this series, 1/x^2 also a fractal in the limit when x->infinity?

As far as I understand it is a series considering the progressive movement of the number.

When you are dealing with something that is so small it cannot be measured:

1) Movement is implied as measurement requires localizes the quantity in a fixed position, hence a series is inevitable.
2) A high set of complex relations which cannot be fathomed except approximately.

The minute nature of the size requires observing x/y with y equaling an impossibly high number which cannot be observed. Looking at a number as merely a set of relations (such as 2 = 1+1, 3-1, 4-2, etc.) the higher the number the higher the set of relations where one infinity becomes quite larger than another. In these respects this high set of relations which composes the number ("y" in this case) in effect makes it in non-mathematical terms "dense".

[wtf]

He of course said that (please notice the bold part) :" In mathematics, infinitesimals are things so small that there is no way to measure them. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were quantitatively small.[1] The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a sequence. Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" means "extremely small". To give it a meaning, it usually must be compared to another infinitesimal object in the same context (as in a derivative). Infinitely many infinitesimals are summed to produce an integral."

I hope it's clear by now that the entire quote, including the bolded part, is from the author of the Wiki article and has nothing to do with anything Leibniz might have said.

[Eodnhoj7]

He of course said that (please notice the bold part) :" In mathematics, infinitesimals are things so small that there is no way to measure them. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were quantitatively small.[1] The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a sequence. Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" means "extremely small". To give it a meaning, it usually must be compared to another infinitesimal object in the same context (as in a derivative). Infinitely many infinitesimals are summed to produce an integral."

I hope it's clear by now that the entire quote, including the bolded part, is from the author of the Wiki article and has nothing to do with anything Leibniz might have said.

And what differs this from what Leibniz "might have said"?