# Are Infinitesimals Really Numbers?

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Yes, I was wrong. Arfa was correct. Infinitesimals approach infinity they are finite.
According to wiki, they approach zero (value).

uestion:
Do infinitesimals necessarily exist?

uestion:
Do infinitesimals necessarily exist?
Can anything that successfully predicts anything be said to "not exist"? Does function trump form?

According to wiki, they approach zero (value).
The 'size' of the infinitesimals approaches zero as the 'number' of infinitesimals approach infinity.

$$\int \limits_a^b f(x)~dx = \lim_{n \rightarrow \infty }\sum \limits_{i\rightarrow }^n f(x_i)\Delta x$$

uestion:
Do infinitesimals necessarily exist?
To integrate the area under a curve you would use infinitesimals. Imagine some sort of curve on a graph and you want to find the area under the curve. You can place a square under the curve to approximate it but that is not very good. It would be better to put a series of rectangles under the curve and add them up to approximate the area under the curve. The narrower the rectangles are the closer they will approximate the area under the curve. As the number of rectangles increases; the closer you get to to the true area. As the rectangles become infinitesimally small and the number of the rectangles approaches infinity the area calculated becomes an accurate assessment of the area under the curve.

Do infinitesimally small rectangles exist. I guess not - but so what?

To integrate the area under a curve you would use infinitesimals. Imagine some sort of curve on a graph and you want to find the area under the curve. You can place a square under the curve to approximate it but that is not very good. It would be better to put a series of rectangles under the curve and add them up to approximate the area under the curve. The narrower the ectangles are the closer they will approximate the area under the curve. As the number of rectangles increases; the closer you get to to the true area. As the rectangles become infinitesimally small and the number of the rectangles approaches infinity the area calculated becomes an accurate assessment of the area under the curve.

Do infinitesimally small rectangles exist. I guess not - but so what?
According to Loll, infinitesimal fractals (triangles) do exist, perhaps even in the abstract.

Can anything that successfully predicts anything be said to "not exist"? Does function trump form?
IMO, functions trumps form. A function can produce many .forms , but a form can produce only a limited number of functions?
Perhaps an example might be the fractal function, which can produce an unlimited number of forms
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A function can produce many .forms , but a form can produce only a limited number of functions?
Well, I am not sure I understand what is meant by "produces" or "trumps", but you are correct is a sense (though I suspect inadvertently).

Suppose the domain of polynomial forms over the integers mod any prime $$\mathbb{Z}_p$$. Then the polynomial forms $$f_1(x)= x^p-x$$ and $$f_2(x)=0$$ determine the same function, namely $$f(x)=0$$

Well, I am not sure I understand what is meant by "produces" or "trumps", but you are correct is a sense (though I suspect inadvertently).
Thank you, if I understand an aspect of a priori causality.I'm a happy guy...

p.s. is *function* possibly tangently related to the fundamental law of *necessity and sufficiency*?

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Suppose the domain of polynomial forms over the integers mod any prime ZpZp\mathbb{Z}_p. Then the polynomial forms f1(x)=xp−xf1(x)=xp−xf_1(x)= x^p-x and f2(x)=0f2(x)=0f_2(x)=0 determine the same function, namely f(x)=0
I cannot comment on the symbolic maths, but IMO, forms are expressions of mathematical functional patterns, as Roger Antonsen demonstrated with his example of two connecting rotating arms turning at a 4/3 rate, creates a remarkably beautiful form, the image of the number 4/3

* The standard reals are complete because they aren't too restrictive, and they aren't too permissive. They standard reals are "just right," like Goldilocks.
Can one drawa comparison to the law of *neccessity and sufficiency*?

Can one drawa comparison to the law of *neccessity and sufficiency*?

I know what necessary and sufficient conditions are, but I couldn't Google any such law. Can you say what it means?

I just think it's a little ironic/odd/interesting that the standard reals are complete, while the two common alternative models of the reals are incomplete by virtue of having too few or too many points. I wonder if that has some philosophical significance.

To integrate the area under a curve you would use infinitesimals.

No this is not true. Each rectangle in a Riemann sum has nonzero base. There are no infinitesimals in the real numbers.

Perhaps I should start by defining infinitesimal. An infinitesimal is a number that is greater than $$0$$ but less than $$\frac{1}{n}$$ for any positive integer $$n$$.

It's clear that there are no infinitesimals in the real numbers. If you claim $$\epsilon$$ is infinitesimal, I'll just keep going $$.1, .01, .001, .0001, \dots$$ till I find an $$n$$ such that $$0 < \frac{1}{n} < \epsilon$$, falsifying the claim that $$\epsilon$$ is infinitimal.

When we form a Riemann sum, we partition the $$x$$-axis into a finite collection of subintervals, each one finite, and form the sum. If the collection of all such sums has a limit, we call that sum the value of the integral.

No infinitesimals are involved in this process, nor could they be, since there are no infinitesimals in the real numbers.

The notation $$dx$$ is not an infinitesimal. It is simply a notation telling you which variable you're integrating with respect to. In more advanced math such as differential geometry, they give formal meaning to $$dx$$ and other differential forms, as they're called, but they are not infinitesimals.

It's true that many students come away from calculus class thinking that $$dx$$ is an infinitesimal, but it is not. The historical achievement of making calculus logically rigorous involved banishing infinitesimals from math and replacing them with limits. Instead of talking about "infinitely small" we talk about "arbitrarily small" and this makes all the difference.

It's true that there are alternative number systems such as the hyperreals and the surreals that contain infinitesimals, but these systems add no clarity to calculus and have their own logical difficulties, such as being incomplete and requiring more set theoretic assumptions than do the standard reals. They're interesting to study in their own right, but they do not bear on the question of whether there are infinitesimals in the real numbers.

There are no infinitesimals in the real numbers.

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uestion:
Do infinitesimals necessarily exist?

Infinitesimals do not exist in the real number system.

Whether any mathematical entities at all may be said to exist depends on what you mean by exist. Does the number $$3$$ exist? Does $$-\sqrt{\pi}$$ exist? How about a noncomputable number that represents a point on the number line but that has no finite-length description? Does it exist? These are questions of philosophy.

But it is a mathematical truth that there are no infinitesimals in the real number system.

The 'size' of the infinitesimals approaches zero as the 'number' of infinitesimals approach infinity.

$$\int \limits_a^b f(x)~dx = \lim_{n \rightarrow \infty }\sum \limits_{i\rightarrow }^n f(x_i)\Delta x$$

No this is mathematically wrong. There are no infinitesimals involved. Please consult any book on real analysis and don't rely on vague misunderstandings picked up in calculus class. Not your fault personally, the teaching of calculus is geared toward practical applications and not to theory, so many students come away with these misconceptions.

No this is mathematically wrong. There are no infinitesimals involved. Please consult any book on real analysis and don't rely on vague misunderstandings picked up in calculus class. Not your fault personally, the teaching of calculus is geared toward practical applications and not to theory, so many students come away with these misconceptions.
I disagree.

I disagree.

With the fact that calculus doesn't teach theory? Or that students come out of calculus class confused? Or that there are no infinitesimals in the reals? I think if you supply some more details, there might be an interesting discussion. Of course my opinions about math pedagogy are my own; but my statements about Riemann integrals and infinitesimals are mathematically correct.

I know what necessary and sufficient conditions are, but I couldn't Google any such law.
Perhaps my use of the term "law" is misleading.
In logic,.... necessity and sufficiency are implicational relationships between statements (abstract perspectives or values), The assertion that one statement is a necessary and sufficient condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either simultaneously true or simultaneously false
(Highlght mine).
IMO, this relational statement holds true in mathematics also.
I just think it's a little ironic/odd/interesting that the standard reals are complete, while the two common alternative models of the reals are incomplete by virtue of having too few or too many points. I wonder if that has some philosophical significance.
Perhaps we have no numerical symbols for values which are unmeasurable to us. Numbers are invented by humans from observation of natural mathematical functions. and values (the universe uses no numbers, it uses latent and dynamic values, potentials). Sometimes we can infer the truth of an *equation* by applying the logic of *necessity and sufficiency*.
Consider the number 4/3 = 1.333333333......in decimal math can also be wxpressed as 1.010101010101,,,, in binary form, yet the value remains the samefrom this *perspective*
Is that not the definition of *axioms*?

The interesting part is that an open number (value) as 1.3333..........can yield a *closed form*

But IMO, logic can resolve a lot of mathematical and observational problems/ Philosopjy gives logical (abstract) perspectives and implicatons to our understanding of the physical world.

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Perhaps we have no numerical symbols for values which are unmeasurable to us.

Yes that's true. There are real numbers for which we can not have any symbols or descriptions or ways of specifying them.

This is a bit off-topic from infinitesimals. There are the non-computable numbers. These are real numbers that can not be described by any algorithm or finite-length string of symbols. There are many noncomputable real numbers. If you pick a random real, it will almost certainly be noncomputable. It's a little mysterious.

By the way "measurable" is a technical term in math that means something else, so best to avoid it.

Numbers are invented by humans from observation of natural mathematical functions. and values (the universe uses no numbers, it uses latent and dynamic values, potentials). Sometimes we can infer the truth of an *equation* by applying the logic of *necessity and sufficiency*.
Consider the number 4/3 = 1.333333333......in decimal math can also be wxpressed as 1.010101010101,,,, in binary form, yet the value remains the samefrom this *perspective*
Is that not the definition of *axioms*?

If I'm understanding you, you're making the distinction between a number and any of its many possible representations. For example 2 + 2, 4, and 10 (base 4) are different expressions that both point to the same number. The number itself is an abstraction.

The interesting part is that an open number (value) as 1.3333..........can yield a *closed form*

Yes in this case that's true. But most real numbers don't have closed-form representations. All the familiar numbers we use do have closed-form representations though. For example the digits of pi or sqrt(2) can be cranked out by a computer program, and a program is a finite string of symbols. We can think of a program as a closed form for the number whose digits it cranks out. But since there are fewer programs than real numbers, there are real numbers without closed forms.

But IMO, logic can resolve a lot of mathematical and observational problems/ Philosopjy gives logical (abstract) perspectives and implicatons to our understanding of the physical world.

To be fair, nothing in this thread is about the physical world. We're talking about the mathematical abstraction of the real numbers. As far as we know, real numbers can't be instantiated in the physical world because it takes an infinite amount of information to specify most real numbers.

It's always important to distinguish math from physics.

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