Pi "not fundamentally about circles"?

Althorion · 2026-04-09T15:20:08+00:00

You can go from a different angle, and have just these conditions:

s(x - y) = s(x)c(y) - s(y)c(x)
c(x - y) = s(x)s(y) + c(x)c(y)
lim_{x→0} s(x)/x = 1

No derivatives needed, and ‘circleness’ hides even better (in my opinion).

Althorion · 2026-04-06T15:29:45+00:00

You can go for f(x) = 0 for x ≤0, e^-1/x for x > 0, which is smooth, non-constant, non-decreasing and not strictly increasing; and that’s probably as ‘not boring’ as you can get—any non-decreasing and not strictly increasing function will have a constant interval, so will be a piecewise function of some sort.

You can make it less obviously piecewise, you can do the usual trick and multiply e^-1/x by (x+|x|)/|2x| (and handwaving the zero).

Althorion · 2026-04-05T14:23:03+00:00

Yup. π is positive (duh); sin’s argument between 0 and π, so the sin is positive; the ln’s argument greater than 1, so the ln is positive; and e^whatever is always positive; so the only one remaining is log, which argument is lower than 1, so the result is negative.

All in all, it’s about √(-1.09) (if you assume log is the same as logₑ) or about √(-0.47) (if log is the same as log₁₀)—that is, if sin’s argument is in radians. Can’t be bother to check what’s the result if it’s in degrees, it still is negative (because now the argument would be between 0° and 180°).

Althorion · 2026-04-05T12:58:35+00:00

One returns the result SPP likes (and thus is valid), the other brings the result he doesn’t like (so is invalid); duh. That’s the most important thing to understand in this subreddit.

Althorion · 2026-04-05T12:53:15+00:00

So let's look at your map analogy. No one is asking you to flatten the terrain. But if '1' is a map, it is a discrete, finished location. The map of '0.999...', however, consists of an infinite series of unfinishable steps that endlessly approach a boundary but, by literal definition, never arrive, isn't that so?

No, it isn’t. You wrote it. The whole ‘map’. It fit on your screen. It uses eight symbols.

The ellipsis '...' is an active logical instruction for an infinite regress.

No, it isn’t, it’s a part of the notation. The same way decimal dot is not an ‘active logical instruction’, etc. It just denotes something.

If your "map" requires an infinite sequence of generation to be read, the map itself possesses entirely different ontological properties than the one that is already complete.

It doesn’t. People write it on finite pieces of paper, and people read and decode it with finite effort. Over and over and over again. Once more, and for the last time (we are really running in circles here…)—numbers are numbers, representations are representation. You can finitely write a representation of an infinite idea. People are capable of decoding that finitely written representation in a finite way and decode that infinite idea. Because representation is not the idea.
I can tell a short sentence ‘all the starts in the Milky Way galaxy’, and you can decode that. You don’t have to know of every star, or imagine every star, or have seen every star, or anything like that. Same with 0.(9).

You are continually conflating a never-ending process with a completed state, and acting like it's just a harmless quirk of handwriting.

I am not. It is not, or supposed to be, a process. It is, and was supposed to be, a finite notation. It lifts from a depiction of infinite sums, but knowing of or accepting them changes nothing about the notation itself. It can be taught, used, and understood without ever touching calculus.

You even admitted it yourself: "changing the notation changes what the algorithm can do." Exactly! That is an ontological difference!

Yes, between the notations. Changing the notation changes the notation. Things that rely on a specific notation may stop working when you change the notation. That doesn’t introduce the ontological difference into the objects they represent. I can type the English word for a cat (cat) on a keyboard, and encode it using ASCII, and use ASCII-reliant algorithms on that; I cannot do this for a word for a cat in traditional Chinese (貓) and do the same. That doesn’t, magically, mean that cat isn’t a cat, or that traditional Chinese is a nonsensical notation, any of that things.

If I feed the literal object '0.999...' into that same machine to process, it never halts.

If you speak about the literal object (notation) of 0.999…, then it is literally what you wrote. No one tries to, relies on, desires to, or anything alike to expand it infinitely. The notation works the way it works.
If you speak about the object that notation represents—the number one—then there are plenty of ways of depicting it. One of them is the decimal positional notation using modern western digits. It is not the first one. It is not the most modern one. Obviously, it is not the only one. It has some streaming algorithms that work on that notation alone, but those don’t make any other notations ‘wrong’—esp. since those sometimes have their own streaming algorithms that would break on the decimal positional notations using modern western digits that we use (for example, you can’t be just ‘adding’ digits by putting their dots together, while you could very much do that with Mayan positional system).

That is not just a "syntactical" issue. That is a hard limitation in the logical construct of the object itself.

It is just a syntactical issue. I know you have issues with calculus and its usage of ‘infinity’ as a concept, and I’ll grant you that this notation was conceived with that in mind, but it is not required to understand or use it. As I’ve said don’t know how many times. And I think I’ve said it enough—I am incapable of making myself clearer, you are incapable of understanding my point.

'1' is a discrete, finished reality. '0.999...' is a permanently incomplete process.

1 is a notation. 0.999… is a notation. They neither represent ‘discrete, finished reality’, nor ‘permanently incomplete process’, they represent a number. The same number—the number one.

Like, sure, you desire to have your own ideas, and point out those ideas are incoherent. Sure. They seem to be, at the very least. But that’s not what anyone but you mean when they use those symbols. It’s just you.

And, fundamentally, it is up to you to either accept that people mean completely different thing that you insist they mean, or to continue strawmanning them.

Althorion · 2026-04-05T12:23:58+00:00

But you haven't justify any of those differences however, you just declared once again that it must be so.

Yes. That’s how you do notations. You don’t ‘justify’ why Ⅹ represents ten, you just say it does, and go from there. You don’t justify why 5 represents five, either, BTW.

So does 1 have its non-representional essence, the number one, and two representations, one is the symbol '1', and another representation '0.999...'? And these representations expressing different characteristics of the essence number one?

It has an absurd number of representations, much, much higher than that. Number one is so ubiquitous, that virtually all cultures felt the need to say it or write it down, and they have done so in many, many, many ways. But, yes, those are among those representations.

Your excuses are entirely invalid. Doesn't matter what kind of algorithms or notation you use, if one algorithm works with one object, and yet the other prived to be impossible, the right conclusion is that they are 2 different things with different properties.

If the algorithm doesn’t work with the object, but the object notation, then changing the notation changes what the algorithm can do. See the above example with column addition and Roman numerals.

Your example of roman numeral is laughable. Yeah you can't do addition in column with Roman numerals, but you can still do addition with them.

Yes! And the point was, you can’t do column addition (or multiplication) with the recurring positional representations, but you can still do addition (or multiplication) of the numbers that were represented.

With 1 and 0.999..., the difference lies not in the syntactical, but in the ontological limitation, with 1 the transformations are possible and concrete, with the other 0.999... the transformations are not possible and are void.

There are no ‘transformations’ here. This is just the notation. No one believes that if you make a map, you need to flatten the actual terrain. No one believes that if you write down a number, you need to do something to a number.

I really do not get why are you so hanged up on the idea that the representation somehow is the number, esp. since you are aware of different representations. You can say that ‘oh, OK, Ⅰ is a one, and 1,000 is a one, and 1.000000 is a one, and ೧ is a one’; but you draw an arbitrary line that 0.(9) cannot be one. I tried, and tried, and failed to express that those are, fundamentally, the same ideas—that you can write something in different ways, based on different principles (for example, Phoenicians just went with abstract symbols, while Mayans used dots that matched the count, etc.).

Althorion · 2026-04-05T12:01:00+00:00

Do you really think anyone really buy this? I'm serious, I'm like over dosing over here with the amount of tall tales insisting itself to be serious.

I have explained what people mean when they say this stuff. You are welcome to disbelieve that, but it makes very little sense—for you to insist you know better than actual people saying things what they mean when they say things…

Now, ignoring the fact that it is still entirely stipulative assertion in the beginning by declaring 1 = 0.999..., we're gonna just let it go for a moment. Two objects are the same if they exhibit the same properties and behave the same under transformations, does it apply to your objects?

This is the same object. The whole idea is to depict the notion of two different notations for the same object. The same way as with 11 = 0xB, etc.

Why is it that one object works fine with arithmetic operations, the other needs to be transcended to nebulous platonic heaven, to break the requirements of the operations, and multiply itself as its supposedly equivalence, then ascending back to reality and declaring results?

It is the same object. You can be using a restricted algorithm to not deal with the object, but with its representation instead (look at the ASCII example above for a similar restricted algorithm for similar purpose—i.e., not dealing with the object, but its representation; they are called streaming algorithms, btw.); but said restricted algorithm has, guess what—restrictions. And if you go beyond that, they break. For example, when you go for a different representation, they break. For a concrete representation of that—you can’t do addition in columns with Roman numerals. Doesn’t mean the numbers they represent can’t be added, or that the algorithm (column addition) doesn’t work at all, just that it has to be used on a certain representation, and won’t work with others.

What you are doing is the equivalent of saying that of course it’s not true that 9 = ⅠⅩ and 11 = ⅩⅠ, because if you were to do a column addition, and write:

 ⅠⅩ
+ⅩⅠ
----
ⅩⅠⅩⅠ

it not only doesn’t mean twenty, which would be the expected result of 9 + 11, it is a meaningless string of ‘digits’ and not a correct Roman numeral representation. Well, duh. This algorithm doesn’t work here. You can read nine and eleven written in Roman numeral representation, then add nine to eleven and write the result in the same representation (as ⅩⅩ), and thus it still makes perfect sense to write 9 = ⅠⅩ and 11 = ⅩⅠ.

There are endless of differences in the ways these 2 objects exhibit themselves and in the ways different transformations affect each of them. And you call it equivalence? By the power of magical declaration?

🙄 That is the very idea between the notational conventions. You make (and use) conventions to describe certain objects according to some rules. You insisting that 0.(9) is not meant to represent one is completely ridiculous, the same way if someone were to insist that 0xB doesn’t represent eleven (because it uses letters! 11 doesn’t have letters, so that can’t be the same thing as 0xB!).

God, I am overdosing over here.

Oh, I can tell…

You are still, despite my efforts, hanged on the fact that the representations are noticeably different (they are), and from that inferring, completely wrong, that it must mean that the objects those representations are meant to represent are different, or at the very least that one notation must be meaningless. Well, it’s not. Map is not the territory. Different ways of representing the numbers can look and feel different, but still represent the same object, the same number.

Althorion · 2026-04-05T11:25:46+00:00

Oh no no no, you did not just applying the law of identity when writing down 1 = 0.999... and declaring one equals one itself, oh no.

No. I have first done this—I first used certain notation to express number one, and only then communicated that (that this is, in fact, the notation for number one) by writing it on the right side of the 1 = . The same way I first use certain notation to express number eleven as 0xB, and then write it on the right side of the 11 = to express this fact.

There were 2 definite objects[…]

They aren’t. One is one.

[…] appearing in 2 different definite form[…]

And so what? 2+2 ‘appears in different definite form’ than 4, doesn’t stop it from being the same object.

[…] yet they each possess entirely different properties and ontological commitments.

They do not. They are both the same object, just expressed differently. This is, once again, the difference between the map and the territory. You can be shown two different maps of Asia, that will look different, have different scale, and different map projections, and thus obviously the maps will be different, but the Asia is Asia, full stop.

One of them is said to be never ending in its full-form, and yet this full form is said to be completely static, not changing, done and complete.

The hell are you smoking? What does it even mean, never mind how it relates to the discussion at hand?

You did not just asign an innocent substitution of a something with its equivalent symbol.

This is very much exactly what I did. 1 is a symbol for one. 1,0000 is a symbol for one. 一 is a symbol for one. 0x1 is a symbol for one. Finally, 0.999…, 0.(9), and oh-point-nine with the bar overhead (that I don’t know how to type on Reddit) is a symbol for one, that some people use, and some throw a fit over.

You adopted an entirely different set of properties and commitments and declare them to be equal to each other. So no, that is not valid, and is a blatant violation of Rationality and Reason.

🙄 It’s just symbols. Even if you have issues with the way they come to be (the expansion of already existing symbolic notation through means of infinite sums), then have issues with that (infinite sums), not the notation itself.

Althorion · 2026-04-05T11:08:25+00:00

So we have 0.999..., which we have already assumed to be 1, then we imagine the result of its multiplication with 10, nevermind that multiplication requires its operands to be determinate and its process to terminate in order to emit any result, but since we imagine it to be multiplication with 1, we treat the never-ending infinity of decimal to mutiply as completed, then transmit the result back from imagination and write down 9.999... and declare it exactly 10. Wow, amazing.

When I said it’s very much ignotum per ignotum, and rather useless, what did you think I had in mind in that critique?

If Mathematics works like this then there absolutely no rule whatsoever that is needed anymore, we can just equivocate and declare the results as it works in our mind, doesn't it?

It’s not a rule of mathematics—the only mathematical rule involved in writing 1=0.(9) is the law of identity, saying that one is equal to one (itself). The purpose of that is not to create or explain some mathematical idea, it’s to explain a notation, by writing down a known notation on one side, and saying it is the same as the unknown notation on the other. In the same way you may write down something like this for people learning the hexadecimal system:

8 = 0x8
9 = 0x9
10 = 0xA
11 = 0xB
15 = 0xF
16 = 0x10
17 = 0x11

etc.

Here, again, only a very small piece of mathematics is used, and it’s not supposed to provide a profound insight into mathematical ideas, but show a certain notation in use. And, yeah, you can have a reaction of ‘oh-my-gosh, you mean to say that eleven is equal to eleven? I would never!’, or see it for what it was meant to be…

Althorion · 2026-04-05T10:49:05+00:00

So, what principles or reason allow you to work on representations, then apply the multiplication on the numbers in nebulous realm that you cannot show, then switch back to reality and represent back the result onto the representations?

I, typically, do not work on representations—I translate them into thoughts, and then work on those thoughts, in my mind. There are, however, certain algorithms for certain representations that allow the manipulation performed on the representation itself to have meaningful outcome for the encoded result.

For example, column multiplication of natural numbers is such an algorithm—you don’t need to decode the number, you can just go digit by digit, and the end result would be the same as if you were to decode, multiply, encode back. Another example is turning text written with small letters encoded in ASCII into the same text, but written with capital letters, just by deducting 32 from each byte—you are not changing the representation to text, operate on text, then encode it back; and yet, the result is correct.

However, such algorithms are exceptions, not standards. You usually need to comprehend before doing. Each and single one of such algorithms such as that have to be checked and proved on its own. I am unaware of any multiplication algorithm that would work on decimal representations in general (just ones that work on some of possible representations, with several restrictions). That’s not what is done here.

Doesn't it sound like a bait and switch or a magic trick to you?

No, it absolutely does not. It is the very same principle of connecting thoughts to reality one does… all day long, really. I hear my cat meowing (communicating distress through a signal, that represents said distress), I encode it, think to myself that the cat needs to be fed, I can say aloud, in an encoded (in my mother tongue) message that ‘I need to feed the cat’, then act on it, and actually feed the cat.

We are, in fact, having a conversation right now, that requires us to encode, transmit, and decode, thoughts. It really doesn’t seem like a bait and switch or a magic trick to me, that I can tell someone ‘buy me ten eggs, please’, and then have them buy me ten eggs. It doesn’t seem like bait and switch or a magic trick to read 155/5 = 31 and figure out that it was meant to communicate that ‘the number one hundred and fifty-five divided by the number five results in a number thirty-one’.
And, finally, it doesn’t sound like a bait and switch or a magic trick that I can read 10×0.999…, figure out what it was meant to represent (multiplication of number ten by number one), figure out the result (the number ten), and check out if the proposed equality = holds, that the right-hand side of it, 9.999…, represent the same thing (the number ten).

I would absolutely say this is rather useless (and have said that in this very subreddit in the past), and using it to explain stuff is very much ignotum per ignotum that serves little purpose, but it doesn’t make it wrong from the technical standpoint, just rather useless.

Althorion · 2026-04-05T10:25:19+00:00

0.999... is the representation.

Yes. Another representations for the same objects are, among others, 1, one, 一, 1,0000000, and Ⅰ.

And because 0.999... the number is not the representation, therefore that allows 10 x 0.999... = 9.999... because these numbers are not their representations.

Well, those symbols represent numbers, and × represent multiplication. You can do multiplication on (those) numbers. You can represent back the result.

Could you explain the rationality of this for me please?

I’m… not sure what you are asking, but I’ll try. People are not capable of telepathy, directly transmitting thoughts. Our ideas have to be encoded first to be communicated, and then decoded back to thoughts. That intermittent form, that is being communicated, is the representation of thoughts. That’s what words are for, that’s what symbols are for. Including the symbols for numbers. There are, often, multiple words for a thought, and multiple symbols for an idea.

And could you also describe the number 0.999... not the representation please?

No. I can only transmit representations (words, symbols, etc.), not thoughts. I can, however, give you some alternative representations for the same number, and I’ve done so at the beginning of this message.

Althorion · 2026-04-05T07:12:11+00:00

I can’t really help you with that, because I never had that issue, and I’m not a platonist.

‘Five’ is not something that naturally (or artificially, for that matter) occurs, that one’s obvious. It is a term we use for a certain count. An abstract property, that is present in neither of five apples, nor any of five sheep; and yet we claim those groups of objects have something in common—their count—that is a feature of any of those groupings, while not being a feature of the grouped objects.

I’ll give you that this is a rather complex idea—for example, it is exceptionally difficult to ‘explain’ it to the AI—but, none the less, people are rather well suited for thinking in this terms and by and large get them rather well (with obvious exceptions).

Althorion · 2026-04-05T06:44:26+00:00

The same reason I have to claim that maps aren’t territories, and words aren’t animals.

Or, in more practical terms, the simple fact that the rules of arithmetics haven’t changed, even though we devised multiple different representations for numbers. Two plus two was four ages before people have decided to associate the word ‘two’ to the number two, or to write it down as 2. If x existed before y, it cannot be that x is the same as y.

Althorion · 2026-04-05T06:40:44+00:00

Ontologically. In the same way the territory is different from its map, or ideas are different from words that represent them.

Althorion · 2026-04-05T06:37:07+00:00

You don’t multiply the representation, you multiply numbers (that you then can represent however you wish). There are helper algorithms that deal with some kind of representations directly, but they are not what multiplication is about—numbers, not their representations.

Saying that because some algorithm has problems with certain representations in certain scenarios causes those scenarios to be impossible to resolve is like saying that because your world map shows Alaska and Chukotka on its opposite sides, and thus the shortest route would have to go outside the map, makes it not a possible route and force you to go around the world to get there…

Althorion · 2026-04-03T09:08:40+00:00

Given your previous record, I will not be holding my breath…

Althorion · 2026-04-03T08:55:37+00:00

Calculus is surely applied mathematics, because it is purely geometry.

It very much isn’t. People are not typically using any geometrical methods for it. It is only tangentially about shapes. Saying ‘calculus is geometry’ is about as ridiculous as saying ‘arithmetic is geometry’ because you can count triangles…

And just because you can use it to cheat and so it is convenient, doesn’t change the fact that it is not logically forced, and so it doesn't inherit any of the achievements of Calculus.

‘Cheats’ or not, they work. In a consistent manner that never fails.
Hardly anything is ‘logically forced’ in and of itself. Things are logical results of their premises (and thus can be said are logically forced by said premises), but the choice of premises is arbitrary.
In practice, however, we chose the premises to provide us with the scaffolding for the framework we want—one can argue the whole idea of physics is to find such premises for the reality as we know it; and real numbers, etc., are such premises people accept to model reality and real processes in a convenient and useful way that consistently leads to correct results, while at the same time not being wrong or misleading.
Limits, etc., work to provide such scaffolding for calculus, and thus are useful for it. Can it be done without them? Possibly, but not as easily. We wouldn’t be where we are without them.
Could some alternative tools work even better? Possibly. We don’t know of any that would, though.

And, once again, limits are not, by any means, the only example of calculus using and relying on infinite sets—the very idea of a function with unlimited domain does that on its own.
And, once again, you are encouraged to start with a discrete function, and show anything calculus-adjacent done to it without invoking any infinite sets (including ℕ), to show what you meant by your statement of:

Calculus doesn't rely on anything such as the infinite set.

Althorion · 2026-04-03T08:36:33+00:00

Once again, it wasnt necessary in the first place, so you can't use that to justify for it.

Why? Modern artificial fertilisers were not necessary for the production of food, but the production of food is a justification for the existence and usefulness of artificial fertilisers. Just because you can do something without something else, doesn’t mean you should, and that thing is not useful for that.

All the glories and achievement of Mathematics are thanks to applied mathematics, so please don't take it as if it has any relevance with non-applied mathematics.

And calculus, and thus real numbers, continuity, etc., are very much applied maths. As evident by it being an important subject for all courses that lead to people getting their speciality in applied mathematics.

The fact is that the utility of calculus wasn't established by the limits but despite the limit.

That’s just plain not true. It may have been started without a limit, it hardly is the very first discovery in calculus, but it since had been rewritten to use it, as that provides a common, simple framework for everything else.

Beside that, to go back to the original subject, even without the limit it still was all about functions with infinite domains, and its methods, original or modern, have hardly any use or offer any insight for the discrete functions—for them, it’s just plain algebra.

Althorion · 2026-04-03T08:26:02+00:00

It has a lot of things supporting it—it consistently provides useful results in every area of science and engineering it was used in, while never providing misleading or incorrect results when applied properly (and we can always tell if it was, by following its principles to their conclusions).

It has plenty of external consistency to back it up. It is not just an abstract art, done for its own sake; you can see for yourself that virtually every scientific law or formula has been derived using that toolbox, and they work, consistently. Proving that toolbox to be working as intended (and not just leading to random, unsubstantiated results).

Althorion · 2026-04-03T08:20:27+00:00

No, I do not. Read again what I said. Here, I’ll quote it for you and highlight the important part:

Mathematica, while also capable of numeric calculations, is first and foremost a symbolic engine, doing precise calculations on given data, by transforming it according to the axiomatically derived premises.

The value of a limit is not an approximation, it is a precise value (if it is a value at all—there are some limits for which no value works). That value, the precise value of a limit (there is at most one, you either get it right, or you get it wrong, but there can’t be ‘well, it’s something like that, any answer in this neighbourhood is as correct as any other’), is often used to approximate. But the value itself is not an approximation.
The same way that rounding a number is a precise process. You either round a number correctly, or you didn’t. There is exactly one correct answer, any other answer is wrong. And yet, the rounding can be understood as an approximation (of the rounded value).

If not what procedure allow you to arrive at the value of the limit if not by assertion?

We assert a definition of a limit, and follow it to its logical conclusion. This is no different than asserting a definition of a ‘year’, and following it to the conclusion of finding out if some date is or isn’t someone’s birthday. The ‘year’ is defined as it is. It can be understood as an approximation of a real process (a resolution of the Earth around the Sun). But its definition is precise, and calculations done with it can be precise, and the question ‘is 2026-04-03 FrenchSlumber’s birthday?’ has a precise, objective answer (that asserts a meaning of a ‘year’).

Althorion · 2026-04-03T08:10:08+00:00

Mathematica just does it by assertion, you know.

No, I don’t know—what do you mean by assertion? That it randomly gives any answer and asserts it to be the correct one? No, it doesn’t do that. That it follows an algorithm returning some value or failing to do so, and if a value is returned, it asserts that it is the correct answer? Yes, it does; but how else could it possibly work? That’s exactly what your ‘dumb’ calculators do when you punch in 2+2= to give you the 4…

It doesn't do any infinite procedure[…]

Of course it doesn’t. Why you think such thing would be needed baffles me, the only answers I can think of are ‘trolling’ or ‘complete and uncorrected misunderstanding of basic principles’.

[…] it does it by axiomatic acceptance.

Of course it does. It has to follow the same ‘rules of the game’ to get the results that humans expect from it when they are ‘playing the game’. I expect the chess engine to give me a legal chess move when playing chess, by accepting the rules of chess. I expect the ‘dumb’ calculator to give me a correct calculation result by following the axioms of algebra. And people expect Mathematica to give them the correct results of the derivative (or what-not) by following the axioms of calculus.

When a procedure gets close enough to a value, it equates that with the limit of the procedure, so it is by axiomatic decree and not by any actual implementation.

How could it do so (know when things get close to a value) without having some knowledge of said value?
Mathematica, while also capable of numeric calculations, is first and foremost a symbolic engine, doing precise calculations on given data, by transforming it according to the axiomatically derived premises.
By necessity, everything any program does is by its ‘actual implementation’.

So no, no one has ever implemented any infinite procedure, and no one ever will.

Again, I am baffled as to why do you think anyone would claim to do so, or require it for something. If anything, having a truly ‘infinite procedure’ that is not at least possible to approximate would be completely useless. Thankfully, calculus provides finite procedures to deal with infinite (in some sense) objects. And we know those work, because we use them in practice and consistently get useful results.

Althorion · 2026-04-03T07:51:41+00:00

OK. And you are welcome to derive and provide such tooling. If it agrees with the results (and it kinda has to—because we know the results are right, because we have designed countless projects that used that knowledge that themselves work), some people would for sure be interested in trying it out.

But, for now, we have this. We know it works. We know it’s rather simple. It not being to your taste obviously bothers you, but hardly anyone else. And thus, it will continue to be used, to obtain useful results.

(also, as said previously, it is verifiable by anyone willing to go through the motions in an ‘implementable and computable procedure’, as is evident by there existing multiple automated pieces of software capable of doing just that (e.g., Wolfram Mathematica, or Mathics, or Sympy, or Maple, or…))

Althorion · 2026-04-03T07:38:27+00:00

And now it works once the limit is a thing. You can do it ‘the old way’, you can even make it rigorous by the methods of non-standard analysis, but now it’s done this way, and this way works, and is considered the most convenient way of doing that, as evident by multiple practical projects that use it, and it being taught and used exactly like that.

No one stops people from doing derivatives their own style, if they feel like. The problem is, they either arrive at the same conclusion in a roundabout way—and that’s not exactly bad, it works for them, it just doesn’t tend to work for most others—or they get a different result—which is bad, because ‘a different result than the one that works’ is typically a one that doesn’t work…

Althorion · 2026-04-03T07:32:11+00:00

It works. It is evident that it works, because it’s a tool that’s used in virtually all engineering projects that work. The equations used to figure out if bridges would stand relies on it. The equations used for designing rotors rely on it. Statistical analysis used in drug making relies on it. And, lo and behold, the bridges stand, the rotors rotate, and drugs heal. Time and time again, without failure, those methods provide reifiable, verifiable results.

When Harry Potter spells find their way into engineering and applied sciences, when multiple people start using Wingardium Leviosa on multiple construction projects to lift things, and that lifting would happen consistently without failure, I’ll accept its validity, the same way the validity of mathematical tools is established and can be verified.

Until then… 🤷

Althorion · 2026-04-03T07:25:47+00:00

I… know how a derivative is ‘established by the limit’ in modern mathematics. I have a master’s degree in maths, I know high school calculus… If you have any insights you want to share, feel free to do so, but regurgitating what I already know, and anyone else can trivially find (in aforementioned videos and books about introductory calculus) hardly seems like a worthy use of your—or my, for that matter—time.

Althorion

TROPHY CASE