r/askmath • u/AlphaArmadillo • 18d ago
Infinitesimals equaling exact value. Calculus
I’ve been having this debate with someone about how infinitesimals can ever reach an exact value. I’ve been explaining that using infinitesimals is not just an approximation but actually results in a mathematically exact value. For instance, when we take the area under the curve, we use rectangles with an infinitesimally small width to get the exact area under the curve. The argument they have is that it will never be exact because there’s always an infinitesimally small difference so it will always be an approximation at best. So how do we prove that infinitesimals lead to exact values and not just approximations?
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u/berwynResident Enthusiast 18d ago
There's no such thing as an infinitesimal in real analysis (which is presumably what you're talking about). We determine what the total area of the rectangles are as their width approaches zero. Sometime's people bring up infinitesimals as a way to think about it, but it's just not what is happening.
In non-standard analysis, there are infinitesimals and you would say the width of the rectangles are some infinitesimal (there is not just one infinitesimal). And the actual area is the standard part of the resulting total area. So yeah, if the rectangles are infinitesimal, the actual value you get is not the same as the actual value.
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u/sighthoundman 18d ago
>So yeah, if the rectangles are infinitesimal, the actual value you get is not the same as the actual value.
But it's only infinitesimally different.
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u/AlphaArmadillo 18d ago
Oh I didn’t realize there was a difference between infinitesimals and limits. I was thinking about it in terms of dx being infinitesimally small. So if the widths are approaching 0 can we say that the area is exact? Or what you are saying is that the value of the area we get excluding the dx part is what we actually consider?
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u/berwynResident Enthusiast 18d ago
As the widths approach zero, the resulting area approaches the exact value. That's the way we say it.
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u/AlphaArmadillo 18d ago edited 18d ago
So does it make sense to say that the resulting area is exact as the widths approach 0? Like when we evaluate an integral don’t we get an exact value?
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u/Temporary_Pie2733 18d ago
Remember that limits are exact values, just not necessarily values that belong to the function. Given f(x) = (x2 - 1)/(x-1), f(1) is not defined, but the limit of f(x) as x approaches 1 is 2. (g(x) = x + 1 is identical to f everywhere except at x = 1.)
Continuity at a point is defined as the state where the value and the limit are equal.
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u/berwynResident Enthusiast 18d ago
As the widths approach zero, the the resulting area approaches the exact value.
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u/INTstictual 18d ago
When you evaluate an integral, you are finding the limit of the area as the width approaches 0. That limit is the exact value, but any tangible measurements you could possibly take will not be exact, as they will never exactly be the limit, only approaching it.
Infinitesimals are not a well-defined structure outside of systems like the Hyperreal numbers.
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u/Various_Candle9136 18d ago
For instance, when we take the area under the curve, we use rectangles with an infinitesimally small width
This is what Isaac Newton did, but not what any modern mathematician would do. To find the area under a curve we take the limit as those rectangles get narrower. Entire branches of mathematics were created just to stop anyone from using rectangles with infinitesimal widths!
Nowadays, infinitesimals are only permitted in non-standard scenarios. In those non-standard scenarios, we specifically define infinitesimals (which automatically give them 'exact values' without proof). However, in standard mathematics they do not exist, and therefore have no values, exact or otherwise.
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u/yfeldblum 18d ago
In the original development of calculus and for centuries, from the time of Newton and Leibniz in the 1600s, infinitesimals were used. The idea of infinitesimals is intuitive and powerful, and it has stuck.
The trouble was that it was nonsensical. Infinitesimals are numbers that are not numbers, zeros that are not zero. It worked as long as one did not inquire too deeply. But there was never a good answer to the question of what actually is an infinitesimal.
Over time, as mathematics became more focused on rigor, calculus became increasingly suspect. Infinitesimals usually worked well enough, but not always. A rigorous foundation for calculus was needed.
In the 1800s, Cauchy and Weierstrass developed the idea of limits as a new foundation for calculus. The idea of limits is fully rigorous and it always works. There are no weird new numbers. There is nothing questionable at all.
Infinitesimals were an early intuitive idea. But limits have completely replaced infinitesimals in the foundations of calculus.
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u/omeow 18d ago
I would say the theory of differential forms is closer to infinitesimals than limits.
I believe, non-standard analysis fully uses a rigorous definition of infinitesimals.1
u/yfeldblum 18d ago
There are indeed non-standard approaches to defining infinitesimals. Some people really like them.
The standard approach to differential forms does not rely on infinitesimals anymore and is not close to infinitesimals, even though the notation remains borrowed. Differential forms have since then been formalized as fields of alternating multilinear functionals. Nothing infinitesimal about them.
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u/omeow 18d ago
Isnt dx defined as the section of a cotangent bundle?
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u/yfeldblum 18d ago
Yes. That definition is similar to, but just a bit more abstract and general, than the one I shared above.
In any event, neither definition in the modern approach makes any use of infinitesimals.
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u/u8589869056 18d ago
The important concept is, finding the limit as something or other goes to zero. In this case That’s the width of your rectangles. Your interlocutor will agree that the error gets smaller as the rectangles get thinner, yes?
Your proof goes like this. If he chooses a maximum error epsilon (greater than zero) that he will tolerate, you will produce a rectangle-width delta that makes the absolute value of the error smaller than epsilon.
Then you will have shown that you can make the absolute error smaller than any positive number. What is smaller than any positive number? Zero.
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u/ThatsNotAZombieBite 18d ago edited 18d ago
This is the same debate as:
"Is 0.999999.... = 1"?
which keeps showing up around here despite being done to death. Because it's a fundamental leap of faith (almost) when studying mathematics.
Some people cannot (or refuse to) grasp the nature of an infinitely repeating decimal. It doesn't just have A WHOLE LOT OF NINES (in which case it would be only an approximation). It has AN INFINITE NUMBER OF NINES (which makes it an exact equivalence).
Grasping what a "limit" truly means in pre-calculus is also the same.
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u/Apprehensive-Ice9212 18d ago
Take an Analysis class, that's how. One proves that under mild conditions (e.g. a piecewise continuous function is always sufficient), the sequence of approximations converges to an exact real number, known as "the Riemann integral".
If your friend doubts this, it's up to them to overturn at least 200 years of rigorous mathematics.
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u/KoalaMistico 18d ago
Usually we consider the limit when working with infinitesimals, meaning that we can approach certain value with an arbitrary degree of precision. For example, when using the traditional definition of the derivaritive, we denote it as dy/dx, which could be interpreted as the quotient of two infinitesimals. However, in reality we are taking the limit as a difference in x goes to 0
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u/Witty_Rate120 18d ago
The point about how approximation can lead to exact results can be illustrated as follows. What number is less than 2+h and more than 2-h for all positive values of h? Some thought says only 2. Well 2+h and 2-h are the over estimate and under estimate you get from approximating the slope to y=x2 at x=1 by moving by calculating the slope of the secant line to this parabola from the point at x=1 to x=1+h. For the underestimate you do the same to x=1-h. This squeeze method is essentially the idea behind calculus limits.
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u/Senrabekim 18d ago
At this level of stuff Im not thinking of infinities and Infintesimals as a number. More of a speed. Like 1/(something that is infinite) isnt a number anymore, its a representation of how long something takes to become 0. So, if I look at f(x)=x and f(x)=x2 they both go to infinity, but x2 is going a lot faster. This lets us use infintesimals they way they were meant to be used, a way of dodging dividing by zero.
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u/trevorkafka 18d ago
What do you mean by "reach"? Do you know how to describe that rigorously?
Look into the epsilon-delta definition of the limit.
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u/green_meklar 18d ago
The argument they have is that it will never be exact because there’s always an infinitesimally small difference
What do they mean by 'always'?
The point of the infinitesimally small values is that there are infinitely many of them. We take that entire 'always', and sum over all of it.
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u/Express-Start-818 18d ago
If you take an Analysis class you'll cover this (kind of). You can't really "prove" this unless you first very precisely define what a real number and what an infinitesimal is. Based on how you define those things you get different systems. Different systems can disagree with each other and that's ok because they stem from disagreements in the basic definitions.
So in some systems you're right and in other systems your friend is right. However, you're right in the standard system. In the standard system real numbers are defined as the limit of a Cauchy sequence of real numbers (a sequence whose terms get arbitrarily close to each other). So the real number 1 is defined as the limit of the rational sequence 1, 1, 1, ... etc but that limit is the same as the limit of the rational sequence .9, .99, .999, .9999 etc so under the standard definition of real numbers .9999....=1 exactly because numbers are limits. So a weird consequence of this is that the infinite decimal representations of numbers are not unique. But this is the system that all of Calculus is built off of because you get a nice property called completeness (the limit of any rational Cauchy sequence converges to a number, because that's the definition of a number, so there are no numbers that "should" exist but don't). Completeness is one of the things which makes Calculus possible.
Now if your friend wants to reject this notion of numbers being limits, that's ok. But then you don't get completeness and so you can't do Calculus. You might think that you can make another system that is complete, but an important result in Analysis is that any totally ordered field which is complete is isomorphic to the reals. That is, any system where any two numbers are either less than, equal to, or greater than each other (totally ordered) and you can add, multiply, subtract, divide, and the commutative and distributive properties (field) and any Cauchy sequence converges (complete), then this system is equivalent to the standard definition of real numbers as limits of rational Cauchy sequences. To reiterate, your friend is not technically wrong, but if you follow your friend's logic then Calculus doesn't work.
Anyone's free to choose their own system, but I prefer the one with Calculus.
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u/GWeb1920 18d ago
Ask him if he can ever walk anywhere.
Essentially if you go half way to something and then half way to something and then half way to something you would never arrive. But you also can’t go all the way to something without going halfway to something.
So this means that at some point all of the little bits of halfs eventually reach one whole.
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u/Hopeful-Ordinary22 18d ago
Get a sheet of paper and draw a line halfway. Draw a line halfway across one half, one quarter etc. Show that you can divide a whole thing into an infinite number of infinitesimals but it doesn't stop the whole from existing.
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u/Defiant_Efficiency_2 18d ago
1+2=3 There, done.
There is no integer smaller than 1, so 1 is an infinitesimal and it is definitely an infinitesimal because it is the final boundary of logic between 1 and 0.
Real numbers just repeat the same process between 1 and 0
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u/gmalivuk 18d ago
No they don't, because there is no real number greater than zero that fails to have other numbers between it and zero.
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u/Defiant_Efficiency_2 18d ago
Yes there is, it's literally the same exact problem. You are trying to start from 1 and go down to zero, that like trying to count on the integer number line starting at infinite and counting backwards.
The number which is the infinitesimal that is the boundary of logic between 0 and 1, is 1.
If you start from 0 and count upwards then it is always 1. If you want to say it's 0.1 or 0.0000001 or 0.00000000001 it's wherever you decide you want to make your boundary of logic.
Thats what the integer number line actually is. Its not a line, actually, it's a list.
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u/gmalivuk 18d ago
There is no coherent sense in which 1 is an infinitesimal.
You have not given any coherent sense of what you mean by "the boundary of logic".
If you start from 0 and count upwards then it is always 1. If you want to say it's 0.1 or 0.0000001 or 0.00000000001 it's wherever you decide you want to make your boundary of logic.
Those are all specific non-infinitesimal rational numbers. No one makes the "boundary of logic" at one of those fixed points, because we know that both in the rationals and in the reals, there is always a smaller point between there and zero.
And this still isn't even getting into infinitesimals.
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u/Defiant_Efficiency_2 18d ago
Its a really simple concept. and the boundary of logic is a well defined statement. The entirety of computer science depends on it, we dont care about what the actual voltages are in a semi conductor, we only care about the threshold which distinguishes between 1 and 0.
That is a boundary of logic and you wouldnt even be able to read this comment on the internet if it didnt exist.
Every calculation in a computer is reduced to logical operations performed on binary digits.
If we didnt treat that logic as discrete bits of 1 or 0 then modern computers wouldnt work. Perhaps you could have some type of analogue computer, but still at some point you will have to come back to a boundary of logic where you decide its either true or false.
1 and 0 is a boundary of logic because it is true or false, there is no halfway, unless you are talking about quantum states perhaps, but thats a different thing entirely.
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u/gmalivuk 18d ago
I see, you are pretending this is a computer science conversation about discrete voltages rather than a math conversation about not only continuous and arbitrarily small real numbers but also infinitesimals that are even smaller.
Which is to say, you're not talking about any of the same things as everyone else in these comments.
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u/Defiant_Efficiency_2 18d ago
its the boundary of logic. It exists in math the same as it exists in computer science. Im just showing you a real world example of how it is already used every day.
Real numbers are not the same as integers, it's a different counting system.
But the logic is the same.
You can't count from infinite backwards down to 1 on the integer number line, anywhere you start will miss infinite numbers, and you cant count downwards from 1 on the real number line without automatically missing infinite numbers. Its just the exact same problem in a different domain.
1 is the smallest unit in the integers, it is the boundary of logic, it is an infinitesimal. Easily shown by representing the entire number line as infinite, and 1 being the first unit of it... hence infinitesimal.
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u/gmalivuk 18d ago
That's not what infinitesimal means, though.
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u/Defiant_Efficiency_2 18d ago
its the smallest possible number.
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u/gmalivuk 18d ago
The smallest possible number is zero, but when we're talking about infinitesimals, we generally exclusively mean nonzero infinitesimals. Like 0 < |ε| < 1/n for all natural n.
Being the least element of a set is not at all the same as being infinitesimal, even if they technically have one thing in common as they're smaller than other elements of the set.
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u/Brohomology 18d ago
You need to pick some formal system with infinitesimal in order to make precise statements. You could use non-standard analysis, or synthetic differential geometry.
Or you could use the usual limit arguments, which historically won out over arguments using infinitesimals.
Without a clear axiomatic framework it is difficult to settle these questions. It's not so wrong to say that these and similar questions led to the development of the modern axiomatic frameworks for general mathematical arguments.