Calculus 5.2d - Infinitesimals - Newton to Robinson

[@derekowens]

@bp56789 Agreed! If you *intuitively* understand the nature of the infinitesimal, as Leibniz and Newton did, then it makes sense to think of those two as different things. In one case you can ignore the dx, and in the other you can't, and why you can in one place and not in the other should be intuitively obvious!

[@Seofthwa]

Thanks for the explanation. I am studying calculus again with the approach of infinitesimals so as to learn it as the way that Newton and Liebniz approached it. It definitely sees easier (to me) to understand rather than the use of limits.

[@bp56789]

Berkeley's argument mentioned at 7:30 is illogical: dividing multiples of dx by dx is completely different to adding dx, but Berkeley has treated them as the same thing. All your videos are very helpful, thank you.

[@derekowens]

Agreed, it's very reasonable if you understand it intuitively. There are times when you need to treat dx as zero, and times you need to treat it as a non-zero finite quantity. And it is hopefully intuitive when to think of it one way and when the other. From what I understand, Newton and Leibniz had this intuitive understanding of it.

[@ChilloutLibrary]

It's like the property of wave-particle duality in physics

[@MagnusAnand]

This is a fantastic explanation. Bookmarked 😃

[@abdlwahdsa]

You can definitely ignore addition of a tiny number as it wouldnt bring much change to the answer, but you can't ignore dividing/multiplying by it cuz this will substantially change the result! if this seems not so logical, then i dont know what is!!

[@juanramirez-ro5ku]

you make integrals more clear. how does distant and area relate?

[@lishlash3749]

For a rigorous mathematical treatment of infinitesimals, see John Bell's "Invitation to Smooth Infinitesimal Analysis". The key concept is that an infinitesimal is a non-zero quantity whose square is precisely zero. In the video above @ 7:00, this allows you to replace dx^2 with zero, BEFORE you divide by dx (which remains a non-zero divisor). You then have 2xdx/dx = 2x, a precise result which does not require you to treat dx as if it were zero. As a consequence, however, you must discard the Law of Excluded Middle, and understand the profound implications of doing so.

[@derekowens]

Lish Lash, thanks very much for the insightful comment.

[@lishlash3749]

***** See the John Bell link above for the formal proof using infinitesimals. With the square of an infinitesimal defined as zero, derivatives can be calculated precisely, without the need for taking limits.

[@lishlash3749]

***** With mathematics, you don't offer a "justification" you supply proof.

[@lishlash3749]

***** LOL, math isn't a pissing match either. If you're genuinely interested in learning about infinitesimals, here's a direct link to the source: publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf

[@lishlash3749]

***** No, it's your persistence in trolling me that's apish, while your refusal to accept the answer I already provided is merely obtuse.

[@DieTGang]

@Derek Owens So, dy/dt or dz/dx is some nonzero real number, but we don't know what exactly it's equal to. But the product of these numbers must be 40.

[@didles123]

Using chain rule we can say (dy/dt)*(dz/dx) = (dy/dz)*(dz/dt)*(dz/dx) = (dz/dt)*(dy/dz)*(dz/dx) = (dz/dt)(dy/dx). It has nothing to do with infinitesimals or hyperreals.

[@mikeguitar9769]

lim(2x+dx) = st(2x+dx) = 2x. The standard part function is a limit function.

[@derekowens]

@TaterGumfries Excellent. I'll try to take a look. Thanks for the info.

[@mikiallen7733]

Would you help me confirm the conjecture and after watching your video that there some intricate relationship between studying the infinitesimals '' hyper-real numbers'' on one side and the fully developed mathematical work on fractional calculus ? and in case of there any , then from where it all started, also if you can point out some of the real-world applications of fractional calculus I appreciate your comments on this

[@nilsbrageludwigeriksson6154]

dx/dy is relevant in carrying the slope. That is what we use them for, division not addition. Makes perfect sense to ignore or get rid of left over alone dx:es. But by all means keep it as +dx, for transparansy. Many students benefit from this. When I studied math the limit concept confused me, I found an old book on infinitesimal calculus and it all made sense to me.

[@derekowens]

Agreed! The newer approach may be more rigorous, but it is also more formal and tedious. The infinitesimals are more intuitive, and the problems make more sense once the concept is grasped.

[@nilsbrageludwigeriksson6154]

@@derekowens Could you do a hybrid?; the old infinitesimal and when left with +dx you laconically state ...and lim dx-x towards zero and thus remove dx? Otherwise the dx has no determined size, it is only a part ratio and you are kind of free to decide how big you want to make it, preferebly zero

[@TaterGumfries]

Actually, Newton did use limits and explain fluxions in terms of limits. It is not easy to find, but you will find properly cited quotations in Goldblatt's book.

[@BPLearningTV]

One thing to note - in order to keep the notion of dy and dx to be infinitesimals in higher-order derivatives, you have to adopt an alternative notation for higher derivatives. I did a paper on this called "Extending the Algebraic Manipulability of Differentials", and also did a KZbin video on it here: kzbin.info/www/bejne/rWWWY3l-mstnetk

[@philippedescharmes589]

thank you for your course, i'm a professeur, but i rest , (the end of my job) i waq a technology french one, and now i beleive thus i do like knowledge, i've also got a master in philosophy. Merci et votre français est bon ! Philippe

[@DieTGang]

What is wrong here: y = 5x and z = 8t Then: (dy/dx) * (dz/dt) = 5*8 = 40 Since dy/dx and dz/dt are fractions, we can change their denominators: 40 = (dy/dt)*(dz/dx) = 0*0= 0 So: 40= 0 What is wrong here? I would be thankfull if you could explain this...

[@derekowens]

I think I can help. You say that dy/dt * dz/dx = 0*0. But we don't actually know that dy/dt = 0. No relationship between y and t is given, so we know nothing about dy/dy. The same is true for dz/dx. dy/dt and dz/dx are unknown.

[@DerMacDuff]

Why is that thing with the dx so reasonable? I mean, you dont tread it as Zero, you tread it as infitesimal small, like 0,00000000000000000000000000000000000001 , so what is the problem? You divide by 0,00000000000000000000000000000001 and you cancel it later out, because what is the reason that you write a number like 2x + 0,000000000000000000000000001 ? If that thing is there or not doesnt change a thing.

[@namdevchintapalli]

Great - Mr.Derek

[@johnplatko8804]

Great video. Thanks!

[@manicmath3557]

Hello sir. Would u say that hyperreals are the sane as limits. And limits are just infinitesimals in disguise

[@derekowens]

I think I see what you're getting at. When thinking about limits, you are thinking about getting really, really close to some value, but not quite there. An infinitesimal is that small increment, which is really small, arbitrarily close to zero. Infinitesimals, which are infinitely small, and their reciprocals, which are infinitely large, can be called hyper-real numbers, as far as I understand.

[@manicmath3557]

@@derekowens yes. The approach where limits replace infinitesimals is so mych more intuitive to me since i can visuallise something of length infinitesimal compared to visuallising something appeoaching 0. Approaching makes it sound like its not a fixed value and feel like its not a statuc value which is weird But since both accomplish the same task and end up with the same result i like to thibk limits are jist infinitesimals

[@FundamentalGroup]

NSA is really interesting. I was able to prove Tyconoff's thm in only a handful of lines.

[@schrodingcheshirecat]

I've adopted a new approach to viewing infinitesimals I refer to as Nil Space Calculus or observer calculus, in which a nil space refers to the 'interval' between an infinitesimal's 'characteristic' value at arbitrary N or rather 1/N and zero. imagine 2 spheres, one red, one blue. red expanding outward, blue shrinking inward. We intuitively say red expands without bounds, but blue is bounded by zero. Picture blue shrinking inwards without bounds. Normally we imagine watching blue rapidly shrinking out of sight, seemingly slowing down as it approaches zero. But instead, Shrink with blue so that it appears the same size. The universe would appear to be expanding, accelerating outward away from the observer without bounds. You always 'see' what's observed in that space. the value never becoming nil. an ever continuing gulf between the observer and zero You never arrive at zero or infinity, just arbitrarily chosen N or 1/N. there is no absolute infinite limit, only the absolute infinite observer. Allowing the extreme tiny value to be ε. You can always be given a smaller ε/n. 1-ε < 1- ε/n < 1 no penultimate number, just a value at N or N+α when needed you can still discard tiny deltas, such as : Lim (N>>α) --> 1 - α/N --> 1 whichever is practical at the time.

[@lifebreath77]

Like your videos - thanks. Of course infinitesimals, differentials, the limit theorem and hyper-real numbers ALL work, because they are all describing in different ways the same underlying reality.

[@qorilla]

Throughout the whole video, I was waiting for some explanation of the rigorous infinitesimals. Only to find out you don't really know that. Me neither but I thought this would be a good starting point.

[@JB-in4dj]

I will learn both and now I know that infinitesimals is now indeed logically sound it can be employed. HOWEVER, it is not as of yet universally accepted in classrooms so ...

[@luisrojassantana]

congratulations... great explanation

[@sto2779]

What does george berkeley thinks about quantum mechanics... you think hes right overall? I bet most of the best physicists today would disagree with george berkeley's logic simply because there is an explanation for everything at the end of the day which he obviously disagrees.

[@amadeo.dujmovic]

remarkable!

[@erkan808]

You definitely read the book "Calculus made easy" ;)

[@derekowens]

+erkan808 Oh, right, I remember that book by Thompson. I have an early edition that I read a long time ago and enjoyed it very much. I would say, though, that the explanation in these videos leans mostly on what was taught to me by my calculus teacher, Wayne Murrah, when I was a senior in high school. He may have read the same book, though! And it is still a book that I would recommend.

[@gribaahaithem3482]

the problem of ignoring dx is when using mathematics in infinitly small physics objects ...

[@DerMacDuff]

And that 2x is only the linear "approximation".

[@Dismythed]

Correction: Newton's theories did not "oppose Christianity", but rather, like his understanding of the Bible, opposed orthodoxy. There is a massive and unbridgeable gap between Biblical Christianity and modern orthodoxy (Pastoral cosmology; e.g. flat earth*). Science is not in any way opposed to Bibical Christianity except where the atheist scientist is intent upon ignoring any theory that has any chance of pointing towards the possibility of a Creator. * There is not one verse in the Bible that promotes a flat earth or says that the sun and all stellar objects revolve around the earth. Job 26:7-10 accurately describes universal expansion, the composition of space, gravity, clouds holding water and the curvature of the earth, saying: "He stretches out the northern sky over empty space, Suspending the earth upon nothing; He wraps up the waters in his clouds, So that the clouds do not burst under their weight; ... He marks out the horizon on the surface of the waters; He makes a boundary between light and darkness."

[@philippedescharmes589]

deterministic inside epistémologics meanings !

[@DerMacDuff]

hm but i think, it doesnt matter, you treat it all the time as 0.000000000000000000000000000000000000001 never as zero ! I mean, you could replace dx with 0.000000000000000000000000000000000000001 and that changes nothing.^^ but im not an expert in this.

[@DerMacDuff]

or maybe i just dont get it ;D

[@philippedescharmes589]

Sorry it was not the appropriate answer, you are searching origin of science with philosophie and believings, great also ! Philippe!

[@DerMacDuff]

oh and when you treat it as like 0.000000000001 you never have problems, i think.

[@surgeonanuruddha8148]

Well I want to correct one mistake.Newtons Principia was based on divine intervention in universal matters.The reason why Berkeley hated Newton and vice versa ,was that Berkeley was Catholic and Newton being a "protestant".Newton was not Christian actually but was an Arian. He correctly deduced that Christ's early teachings was distorted at Nicea. That is why he was referred to as the infidel. However non standard analysis Rocks.

[@jacoboribilik3253]

fuck limits, infinitesimals are way easier to assimilate.

[@derekowens]

Yes, that's basically the idea. Except I would have phrased it a little differently.

[@jacoboribilik3253]

controversial ideas require controvesial words.

[@phrainkdrums5704]

Don’t talk about infinitesimals if you can’t say how Robinson “proved” they exist. You have to define real numbers before you separate 0.999... and 1. Cauchy defined real numbers in a logical way. Limits are the only logical way to calculus. Let me explain why infinitesimals don’t exist. It’s because infinity is not approachable. For example, 1,000 isn’t any closer to ∞ than 10. If you have unlimited golf balls and I take away 10, you still have unlimited. If you have unlimited golf balls and I take away 1,000, you still have unlimited. Therefore, (∞ - 1,000) = ∞ = (∞ - 10). Therefore, 1,000 isn’t any closer to ∞ than 10. Therefore, 1,000 is 0% closer to ∞ than 10, not an infinitesimal %. If infinitesimals existed, 1,000 would be an infinitesimal % closer to ∞ than 10, not 0%.

[@derekowens]

I hear what you're saying. Newton and Leibniz, though, both did Calculus without any theory of limits. And the thinkers that picked it up after them, such as Bernoulli, did the same. And they got it right, and I believe their intuitive understanding of the matter is valid. Specifically, the example you gave of (∞ - 1000) = ∞ strikes me as similar to Leibniz' example of x + dx = x. If you grasp the nature of the infinitesimal quantities, then it typically becomes clear when to ignore them as essentially zero, and when to treat them as having a meaningful size.

[@phrainkdrums5704]

Derek Owens right. It’s a touchy subject. It’s awesome that you have a video of the old calculus, because a thorough video on it is hard to find. It’s just it would seem quite illogical to have infinitesimals.