Finite versus infinite and number systems | Sociology and Pure Mathematics | N J Wildberger

[@Vikash137]

I think his point basically boils down to infinite decimal expansions don't exist because we can't sit down and physically work it out (since this would require infinite time). But when did the existence of a mathematical object became reliant on it's computability? Can you have a computer or algorithm work out what happiness feels like? Or any other emotions for that matter? Did any number exist before computers or algorithms?

[@elcapitan6126]

The computability criteria is more or less a practicality constraint. One certainly doesn't have to work with this constraint but it is interesting to explore mathematics from this kind of constructivist approach. Pure maths without computability is essentially formal symbol manipulation (I'm not placing value judgement on that though... the ideas are evidently very interesting and compelling to many mathematicians e.g. in abstract algebra)

[@notator]

When I left high-school (late 1960s) we had different exams for what were called "pure" and "applied" maths (the distinction was a bit vague). That was before computer programs started influencing maths' development. It seems to me that "pure" mathematicians should develop whatever they like, regardless of any applications their work might eventually have in the real world. But they shouldn't then complain if the real world starts ignoring them. Keeping in touch with reality is necessary for survival. Pure mathematics is a valuable pursuit in its own right, but I'm firmly on Prof. Wildberger's side of the fence: If there's a pure mathematics that _obviously_ leads to more elegant/efficient models of the real world, then that kind of mathematics should be fostered. Wandering about in a swamp gets us nowhere. On the applied maths front: I think this series is _actually_ aimed at _all_ of us working in disciplines that use mathematics, but whose knowledge of pure mathematics is a bit shaky. Anyone who uses a computer is ultimately using mathematics, so the disciplines include not only sociology, but also physicists, neuroscientists, biologists etc. (who investigate the world as it is) and "artists" (who create new worlds). My own field, music notation, has an inertia that is even greater than the inertia Prof. Wildberger is complaining about. Standard western music notation is stuck firmly in the 19th century. It contains unresolved paradoxes because it still assumes that a temporal ether exists! I think social scientists would be more interested in the _general_ problem of resistance to change than in an analysis of the mathematics they use (mostly statistics, as far as I know). An old classic, still worth reading, is Kung's _Structure_ _of_ _Scientific_ _Revolutions_ (1962). Meanwhile, thanks to this channel, I'm continuing to catch up with the maths I need in order to communicate meaningfully with people in other disciplines. Many thanks, Prof. Wildberger!

[@Re-lx1md]

While I may not have the education to form my own opinion, I love hearing alternative viewpoints and I think that students, especially pure math students, should be exposed to this "side". Thanks for this

[@Re-lx1md]

@Gennady Arshad Notowidigdo I mean I happen to disagree with him about real numbers as well, but I don't think he's selling anything

[@TemplarX2]

I love this channel. You explain everything with such eloquence and clarity.

[@JoelSjogren0]

Hi Norman, I have been thinking a bit about an argument that I think you presented several years ago, about the "p-adic number systems" versus the "real number system", where if I recall correctly (at least this is what stuck for me) the p-adic operations are in a sense "better" (i.e. from a pure mathematics point of view) because they allow truncation, as in "(x * y) (mod p^n) = (x mod p^n) * (y mod p^n)", where the corresponding statement about decimal numbers is false due to carrying going in the reverse direction. But I haven't been able to figure out just how significant this difference is. After all, neither system allows a decision algorithm for exact equality (assuming the numbers are represented as algorithms producing the digits), and the systems have in common the idea of working with "arbitrarily fine finite approximations" even if very ironically the formalization of this finiteness aspect involves yet greater infinities in the form of topologies as subsets of the power set (the notion of "open set" providing (at least that's the idea) a means to express statements to the effect that "if the 'true value' of x is within this set U, then one will be able to observe and thus verify this fact in finite time, completely mechanically, while if on the contrary the true value "actually happens to" lie outside the set then the falsification of membership is not necessarily possible in finite time" - an idea that I learned from the CS department, under the name of "locale theory" or maybe first "domain theory"). Apparently the decimal version of the idea of a real number system yields in this sense a "discontinuous arithmetic" if one takes finite truncations to represent disjoint open sets. One idea for overcoming this particular problem is to grow the size of these intervals by a certain factor such as 50% to ensure that there is some overlap between these open sets. This actually seems to "work" (even for the purpose of establishing the continuity of division "R x (R \ {0}) -> R"), but we are now working with intervals as opposed to idealized "points on the real number line". If I recall correctly this was your conclusion too, that one needs to work with finite intervals. On the other hand it seems like type theory (via its connection with domain theory and locale/topos theory; exemplified in an analogous situation by the idealization in cubical type theory of the homotopy theoretic aspect of the "real unit interval" that enables one to speak somewhat freely about variable "points in the unit interval" while the underlying implementation of cubical type theory takes care of discretizing this, in effect, into statements about "(gluings of cartesian powers of) intervals" thus eliminating the (intuitive and thus useful for reasoning but) unfeasible "pointlike" aspect of the interval variables; perhaps also realized in the work of Paul Taylor on "abstract stone duality", but I'll have to read more about that work to see what it's really about...) could provide a mechanized way of turning convenient language about idealized points into corresponding precise statements about intervals.

[@abdonecbishop]

An educator’s work never ends… well done Norman from your Ontario roots … have listened to some of your concerns about the foundations modern mathematics. For me compactness forces the infinite to have a bounded near zero value in the neighborhood √-1 graphed at point i•(0, 1)

[@fjolsvit]

George B. Thomas takes the approach that statements about infinite sets do not intend that such a set can be exhibited directly. Rather, there is some non-terminating algorithm which enables to make statements about things "going to infinity". The literal meaning of 'infinity' is not having a finish, or termination. So an algorithm which has no stop state is, in that sense, infinite.

[@chosenone2256]

Numbers are formal objects; they don’t need to all actually exist as points in space somewhere. Are there an infinite number of points in space? Maybe not, but I can define a formal object that is infinite by definition.

[@jasonpenick7498]

The reason there is not a major division or debate is because the education system and most of those in the field refuse to participate or shut it down outright, and because those in the field don't seem to want to have conversation with those outside the field who might have valid input, like computer programmers like myself. I've been trying to have a discussion with anyone for years now.

[@figur3itout307]

I loved this discussion as I subscribe to many of your ideas on real numbers and the infinite. I feel what is missing from number systems is their role ‘In Time’. “Real numbers” exist in the ‘real world’ which is a metaphysical idea. Numbers don’t exist physically so they are not restricted by time (or entropy) because they are needed to describe such ideas. Quantities of things do exist in the real world as finite numbers. Numbers that build bigger numbers through time. At any given instance of time we make measurements. Discretized measurements. We can relate these measurements to infinite number ideas because we can see their ‘global’ pattern which ‘transcends’ a physical time barrier and allows us to predict things in the future. Infinite series limits are our way of measuring the boundary of something that is not specifiable on a certain ‘scale’. Just like the Earth appears to have a boundary that one might sail off of and as ships pass over the horizon seem to disappear, the boundary does not actually exist, but measured from a distance it does because our measurement of the ship sailing stops there. Natural numbers are how we measure the concrete. Rational numbers are how we measure ‘scale’ differences of two discrete quantities and these numbers are enumerable in time (ie map to the natural numbers). Infinities occur when we can not measure the relative distance between the discrete quantities in a ratio, when a boundary of sorts has been sailed over. That’s why we use ‘approximations’ with rational for irrational numbers. We can see the pattern but we can not retrieve all of the information in finite time which is why we ‘accept’ decimal values (that are described infinitely) as concrete values. What I think our mind does in the ‘meta mathematical landscape’ is associate the ‘real number irrational’ to the best concrete quantity (rational) is feels comfortable ignoring the error on. Like most people are comfortable with saying Pi = 3.14 bc that’s all they really need it to be, we get comfortable feeling that these abstract things called numbers which can describe concrete discretized quantities are ‘things’ rather than virtual tools. If people really want to understand numbers they must first understand what they are not.

[@hyperduality2838]

Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! Time symmetry is dual to time asymmetry (entropy) -- time duality. The future is dual to the past. Generalization or the infinite (concave) is dual to localization or the finite (convex). Affine geometry or projection from infinity is dual to the finite projection or hyperbolic geometry. All observers have a finite, localized, convex or syntropic perspective. From a convergent, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics. Concepts (numbers) are dual to percepts -- the mind duality of Immanuel Kant. Deductive concepts (mathematics, analytical reasoning) is dual to inductive concepts (physics, synthetic reasoning ) -- Immanuel Kant. Concepts (numbers) are created from deductive and inductive inference.

[@notator]

Much agreed, but I think there are two things that could be made more precise: 1. What do you mean by saying that the "real world" is a "metaphysical idea"? and 2. Which "Time" are you using to define "number systems"? For me, there are two different, legitimate uses of the word "time", so two different but related "number systems": First, there is _experienced_ time (a brain strategy), which we use when defining the counting numbers (integers), and then there is _physical_ time (that is a component of the physical space-time in which we live). We don't experience the "time" component of "space-time", but it can be modelled mathematically. Experienced time can't be subtracted or divided, but we can still, going forwards in time, use the basic mathematical operations to calculate experienced quantities (in the "real world"). Entropy is the _result_ of this inability to make experienced time go backwards, it says nothing about the "time" represented by the "t" variable in (quantum-)physicists' equations. As a species, we have the ability and freedom to develop abstract mathematics that is independent of our experienced world. Pure mathematics is important, of course, but I agree with Prof. Wildberger, that techniques that have only evolved to prop up shaky theories about our experienced world should be retired. More attention should be given to techniques that better model the "real world" (both inside and outside brains).

[@wetyuu]

Any books to learn more about this side of mathematics?

[@rrr00bb1]

It all comes down to whether you can build a machine; that can complete the task. Machines need well-defined instruction sets; for both computing answers and doing symbol manipulation. Without a machine, progress is put into the brains of mathematicians; and most of that progress dies with them. Put mathematics into code libraries, and knowledge truly does accumulate. Infinite iteration (ie: S = 1 + 2 + 4 + ... ) can't be physically realized; but are great for engineering approximations. But infinite recursion CAN be realized. ie: (S = 1 + 2S = (2^n)-1 + (2^n)S). A solution with O(2^n) complexity for large n for time or space isn't a solution if you need actual answers; but perhaps it's ok for proving that certain bad things won't happen. Even with rational arithmetic, you eventually run out of space and time to calculate with integers. I like the idea of changing the math notation until it is well-defined for computers, meaning that a machine can be built to run without human supervision. Standard math notation is full of what we would call usability bugs; where some bits can't be taken literally, in order to avoid wrong answers. And in some cases, where the answers are not wrong; patterns don't match due to notational gaffes (ie: pi vs tau). And in other cases still, the algorithms can't beat brute-force guessing in the search space of algebraic manipulation.

[@rgerk]

What good is a number system if you can't perform arithmethic with it? How do we define root 2 + Pi + e? Sure we can express Pi and e with generalized continued fractions, but how can we sum these? Is there an algorithm for it?

[@whig01]

pi = -i ln -1. As for defining ln and e....

[@ThePallidor]

Root 2, pi, and e are processes, not objects, so they cannot be added or multiplied. They are processes that generate sequences of objects, though, and you can do arithmetic on those objects. 3.14 + 2.72 = 5.86.

[@lanparty2001]

Yes, Bill Gosper developed lazy computation of even infinite stream (i.e. irrational) continued fractions in the 70s, they can be evaluated to arbitrary precision. perl.plover.com/yak/cftalk/ this is slides of talk given that breaks the method down, some more searching will yield the original paper by Gosper

[@hihoktf]

@@lanparty2001 Does it work if you include the axiom of choice?

[@lanparty2001]

​@@hihoktf its an algorithm for exact arithmetic on both rational/finite and irrational/repeating continued fractions. not sure what the axiom of choice has to do in this context

[@hyperduality2838]

Generalization or the infinite (concave) is dual to localization or the finite (convex). Affine geometry or projection from infinity is dual to the finite projection or hyperbolic geometry. All observers have a finite, localized, convex or syntropic perspective. From a convergent, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics. Concepts (numbers) are dual to percepts -- the mind duality of Immanuel Kant. Deductive concepts (mathematics, analytical reasoning) is dual to inductive concepts (physics, synthetic reasoning ) -- Immanuel Kant. Concepts (numbers) are created from deductive and inductive inference.

[@michaelshawn6791]

Why do you spam this? What do these ideas accomplish? Do you need meds?

[@hyperduality2838]

@@michaelshawn6791 Its not spam as there really is a 4th law of thermodynamics! Space is dual to time -- Einstein. In physics you are confined to the 4 dimensions of space/time as outlined by Einstein. To think in 3,5 or 6 dimensions etc. is technically not allowed as the physics demands this! Mathematicians ignore the 4th dimension of time all of the time but this is not allowed in physics -- the physics does not work. "Perpendicularity in hyperbolic geometry is measured in terms of duality" -- Professor Norman J. Wildberger universal hyperbolic geometry. Orthogonality or perpendicularity = duality! Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics. Teleological physics (syntropy) is dual to non-teleological physics (entropy). The electric field is dual to the magnetic field -- Maxwell's equations. Electro is dual to magnetic or photons are dual. Pure energy or light is dual. Duality creates reality!

[@michaelshawn6791]

You believe what you're saying is profound, I'm sorry to break it to you but it's just word salad. You're either a bot or a schizophrenic.

[@davidkeirsey9477]

A really important discussion. One needs to bring into the discussion, the "framework" of the FINITE PREORDER of the 27 FINITE SIMPLE NON-ABELIAN SPORADIC GROUPS, within the context of the 18 FINITE SIMPLE NON-ABELIAN LIE GROUP FAMILIES. Some day Professor Wildberger might see this, but probably not. The understanding of the 15 supersingular primes, supersingular values, and the modular functions of one variable would help him with addressing this conundrum. Building a list of the some important FINITE RATIONAL SEQUENCES (ala Sloan's Integer sequences) would be very fruitful in this enterprise.

[@RoyceFarrell]

thank you it is imperative we have this discussion.

[@manwork6545]

Mankind is not yet ready to have such debate! We need time to improve more and to get wiser. Time will come...

[@patrickwithee7625]

If you don’t believe in infinite sequences, but you do believe in the natural numbers, then what gives? Is not the sequence of all positive integers a(n)=n+1?

[@RichardAlsenz]

The study of space has since Euclid contained an assumption that is proclaimed obvious, but in fact, it is absurd. That absurdity is the source of irrationality. No one has ever seen a point, but only a few realized they are looking at a picture of a point. For those who fail to see this pointlessness, research Magritte's "this is not ??????" There it is displayed in a wide variety. Gauss was aware of this, however, review Bessel's response and it is obvious Bessel did not. Gauss to Bessel Goettingen 9 April 1830 … The ease with which you delved into my views on geometry gives me real joy, given that so few have an open mind for such. My innermost conviction is that the study of space is a priori completely different than the study of magnitudes; our knowledge of the former (space) is missing that complete conviction of necessity (thus of absolute truth) that is characteristic of the latter; we must in humility admit that if number is merely a product of our mind.

[@ThePallidor]

The trouble with basing the critique on what computers can do is that the idea of pure math was always (AFAIK) to use totally non-physical notions, partly on the idea that something discovered in the abstract may turn out to be useful. The epistemological critique is more thoroughgoing and leaves the mainstream no excuses. There is simply no way to imagine things like "infinite sets," so there is no sense using the words. The approach from epistemology forbids pure math in the sense it's currently used.

[@StephenPaulKing]

I disagree. We can make sense of infinite collections, sets, Classes, etc by using mereological tools. The relationships between Whole and Part are, IMHO, the crucial distinction between finite and infinite collections and almost all of the problems occur when mereologies are mixed and not tracked.

[@TheDummbob]

@@StephenPaulKing Could you recommend some text or lecture where one can learn about this distinction between the part and the whole in mathematics/logic? That sounds quite interesting! :)

[@Alkis05]

I would point out that much of the computational problems has to do with modern computers being digital, discreate machines. A lot of the problems of dealing with real numbers were never a problem for analog computers. The same was true of geometric calculations. In such machines, discretization occurs only when reading the results (by making a measurement or sampling a AC current to get a voltage level.

[@Aliusman.781]

I am learning French and I need to say c'est brilliant.

[@patagoncapital7597]

Hi Norman Are you aware of the work of Bill Gosper defining "real numbers" as continued fractions? In his work he claims that both e an pi can be represented as exact continued fractions (given by finite algorithms) and the algebra (e.g. e+pi) is also well defined as another exact finite algorithm producing the resulting continued fraction. I am curious to hear your opinion about his results.

[@njwildberger]

@Patagon Capital Attempts to base real numbers on continued fractions run ultimately into the same kinds of problems having to do with pretending that you can "do an infinite number of things", even though you obviously can't. But the continued fraction approach runs into difficulties in the actual numerical work even with finite decimals. The actual implementation of such an arithmetic is just very unwieldy, even if it is perhaps theoretically pleasant.

[@patagoncapital7597]

@@njwildberger Thanks for the response! I have to look at it more carefuly but it is striking to me that the simple continued fractions of the square roots are periodic and even for pi if we use generalized continued fractions the corresponding series have a simple regular structure, so the algorithmic specification of such "real numbers" is finite. I have to check the algebra more carefully but I guess you are right and the sum of algorithns is another agorithm that is more complex than the previous ones.

[@StephenPaulKing]

Could we use Equivalence Classes to define numbers? For example, the number 1 is defined by the equivalence class of all (possibly infinite!) processes that generate it as a solution.

[@alexcao7502]

A way to define the real numbers is as the set of equivelence classes of rational Cauchy Sequences

[@carly09et]

No ... because oo =/= 00. The counting has to be defined which cannot be done - incompletness problem - it forks for ever !!!

[@johnk8392]

I see two completely different concepts here. One is a number, an immutable specific existing value while the other is a *process* that uses time, soace and energy to do *work* and that it, theoritically at least, finally result in a number like the aforementioned. Those are apples and oranges in my opinion, they cannot be compared and they cannot be used together. E.g., the number 1, have specific value and that's it. The mathematical expression (5-4) also results in the value 1 however the expression MUST be evaluated, it must consume resources to be converted to 1. So no, they are not the same. Fixed values and unevaluated expressions or functions are not the same thing.

[@carly09et]

@@johnk8392 True. It is more general than work and process - that is a very physical view. It is dimensional slippage. An apple is an apple. {for the unit of apple} But one is not one yet 1=1. The question is what is the unit{basis} here??? Mathematics is an art! The first step in mathematics is abstraction - this is the ur use of the axiom of choice - making a model. Ur is before formally defined. Mathematics elegance comes from dimensional slippage - mapping space to space {unbound}. The limit is it is always a model [with a hidden assumption] this gets lost in the meta!!!

[@mustafanawaz243]

Hi Norman, what do you think about using alternative number systems to somewhat tackle the issue of indefinitely repeating decimals. Perhaps we have difficulty with computing square root of two in the decimal system but it would be very easy to write it down in the binary number system. edit: Mistake! I was thinking of 1/2, which is not root 2.

[@Nah_Bohdi]

I use lots of different base systems in my work, including very abstract ones like ratio bases, base fibbonaci and more. What I found was some things become very easy but others become more difficult (or impossible all together) and that there is no system that will make everything easy. No matter what there will be an inability to write something out so the goal is least difficult or other reasons (base-10).

[@spamwithegg]

A representation in binary form does not solve the problem. One can represent a point within the inverall [0,1] with the interval halving procedure "0 for left side of the half interval and 1 for right side" This will result in a sequence with digits 0 and 1, which represents a real number with the interval [0,1], if you accept infinite squences. For example: 0.0101 would be: 0 left side interval [0, 1/2] 1 then right side [1/4, 1/2] 0 left... [1/4, 3/8] 1 right and so on. You can do this with a 2 dim square too. In this case you have 4 digits. 0 bottom left square 1 bottom right square 2 top right square 3 top left square. This is a continous bijection by the way. Both are infinite - say real numbers.

[@peterjansen7929]

My apologies if I am just repeating what others have already written - there are supposedly already 2 replies, but they are both invisible to me! The square root of 2, or indeed ANY natural root of any natural number other than a pure power, can not be expressed as ANY fraction, while ANY given fraction will ultimately be repeating in ANY number system (because in repeatedly dividing one runs out of remainders and encounters the same remainders all over again and again). The fact that roots are either natural numbers or can't be fractions is very easy to show (though school teachers like to present needlessly complicated evidence - itself an interesting sociological problem!): Let p/q (expressed in lowest terms, so that p and q are co-prime) be the k-th root of n. Then (p/q)^k=n, and so p^k=nq^k. As p and q are coprime, there is no prime factor of q on the left side of this equation. Thus there is also no prime factor of q on the right side, so q does not have any prime factors, so q=1 and p itself is the k-th root of n. And that is all the proof we need!

[@ThePallidor]

The problem isn't really a problem in the first place. Simply, "irrational numbers" do not and cannot exist as mathematical objects because the notion is incoherent, but pi, e, sqrt(2), phi, etc. are processes (or sets of processes) that *generate* [rational] numbers.

[@mustafanawaz243]

​@@peterjansen7929 @ Uwe Hennig Thank you pointing out the mistake. I was thinking of 1/2 which is a fraction, and not equal to square root of 2.

[@beatrizkarwai6763]

i do think infinite sequences are important, but they can be difficult to work with. we use numbers to represent concepts which exist in nature, and one example of an infinite sequence is the sequence of partials associated with a particular sound. the harmonic series is the most known example, but objects with a complex shape which isn't dominantly 1-dimensional create different series. so in reality we have such real phenomenon which require infinite series as part of their mathematical description. i think that, instead of avoiding the concept of infinity altogether, we should think about it more carefully. that's why i like to see your perspective on it.

[@ThePallidor]

Well that's always a finite sequence. Rather than "infinite" we simply need to specify process vs. object. Pi is a process (or set of processes), not an object. You carry the process out to as many decimals as you need.

[@rickshafer6688]

All of the analysis books I have are for more exact roundings. Difference applied and pure.

[@santerisatama5409]

"Fractions modelling continuum/continua" is very deep and important foundational observation. Building fractions from natural numbers and/or integers is the wrong way to do it, IMHO. I consider fractions independent from and primary to discrete number theories. I'm trying to learn to think of Stern-Brocot type structures as generating mereological structures of continua; continuous processes.

[@ThePallidor]

You might like John Gabriel's approach, going via Euclid.

[@santerisatama5409]

@@ThePallidor You mean 'New Calculus? I gave a quick look, the guy seems to have rather polemical style, which can sometimes be entertaining. As for Euclid, there's much tension between Archimedes' calculus, and Euclid's 2nd postulate that "Line has no width". Archimedes was kinda saying that "A line can have weight"... these issues go pretty deep into foundational issues. Myself, I'm not especially interested in numbers, but start from a formal language with mereological bent, where numerical fractions arise IMO in pretty interesting way.

[@lanparty2001]

@@santerisatama5409 Bill Gosper did the heavy lifting for defining a computation approach for rational number arithmetic without resorting to fractional interpretations, essentially by conceiving of mappings between rationals as linear transformations/homographic functions encoded by matrices. He used it to tackle exact arithmetic with rationals encoded by continued fractions, but this approach was eventually translated to Stern-Brocot and Calkin-Wilf representations by researchers in the 2000s and onward (the QArith library in Coq is based on this). I would encourage you to look into it.

[@santerisatama5409]

@@lanparty2001 Thanks very much for the link, very interesting. What I'm amazed by is that the formal language expressions I generate contain much more structure than the numerical SB-interpretations; there's kind of a multitude of theories of "rationals", which are highly related and interconnected, but still not "same". BTW the interpretation procedure from formal language to numerical structure utilizes the data types that Norman distinguishes, each once. :)

[@lanparty2001]

@@santerisatama5409 interesting! is there a way we can correspond about this because its something im actively researching atm

[@MisterrLi]

You could re-label the discussion to be "The limitations of real numbers" because, of course they "work" in the sense that they are practically usable and don't lead to contradictions in normal use.

[@kevionrogers2605]

With the knowledge of Plank's constants there is a physical limit to material size, so when dealing with engineering and other physical applications there's no reason to calculate any decimal places beyond that scale.

[@MisterrLi]

@@kevionrogers2605 I was only referring to the real's practical use within math. Sure, they can be used to model nature as well.

[@ThePallidor]

Real numbers are never used at all.

[@walkerjian]

So where in the physical universe does pi = pi? and how much does it equal it? :)

[@kjekelle96]

what about pi and e for example?

[@RooftopDuvet]

Professor Wildberger, have you been following the Geometric Algebra movement at all (particularly PGA and CGA)? I haven't seen you mention it anywhere, and yet it seems to align so well with the principles on your channel. It still uses real numbers, and there are still people squeezing infinitesimals into it, but it's not grounded on them. Seems like a great medium through which that 'debate' you want could occur.

[@dmc2925]

I can’t remember exactly where but he does follow it and mentioned it while explaining Clifford algebras.

[@RooftopDuvet]

@@dmc2925 Ok that makes sense. I'd be surprised if he hadn't mentioned it. There's been some great developments over the last few years though, that Norm's ideas could find a lot of fertile ground in, so I guess I'm still curious as to why he hasn't been more vocal.... I guess there's only so much time in the day though

[@dmc2925]

I found the video! It was a comment on his Math Foundations series, video 69!

[@RooftopDuvet]

@@dmc2925 Haha thanks, that's a very specific moment you've tracked down. The one on polynumbers right? I just stepped through it but I couldn't find anything though... That said, watching bits of that video made me want to go back over Norm's stuff on calculus and polynumbers again. I wasn't even thinking about those videos when I suggested the link to his work with GA, but seeing calculus and polynomials expressed in terms of the binomial expansion has clear connections to pseuoscalars and poincare duality etc, so it'll be great to watch that stuff again with that context in mind. Thanks for the reminder.

[@rossholst5315]

Can you set up a valid arithmetic with just the rational numbers? It would seem that you could not reasonably include all of the rational numbers as this is also an infinite set. It would seem finite numbers such as 100,000,000! while finite is not quickly expanded.

[@rossholst5315]

Also how does this relate to the difference between the arithmetic and geometric mean?

[@rossholst5315]

Finite rational math systems…that is…

[@njwildberger]

At least with rational numbers the arithmetic does not involve an infinite number of operations. Since addition is defined as a binary operation you only need to get your computer to recognise when an expression is a rational number and then to have an algorithm for taking two of these expressions and adding them. The computer does not need to preload all of the rational numbers

[@rossholst5315]

@@njwildberger I am very intrigued by your ideas and approach. I have been troubled by many of the paradoxes or incongruities caused by the continuum. I guess my thoughts are that you don’t need to have an infinite number of steps to make a task impractical to complete. 1 million factorial should be an integer, a computer should be able to calculate this number using a simple algorithm. However the problem becomes the length of digits and the information that needs to be stored, and the potential number of steps. While a computer can make computations much more quickly that I can, there is still a finite amount of time it takes to complete a step. Even if this interval is small, if there are enough of them, the time to complete the task could become so long that it is impractical to complete. But even if my calculator could produce this integer, using the exact number for additional calculations would be unwieldy. The calculator on my phone says 1,000,000! is undefined, yet I know it is an integer, but what specific integer it is I cannot say. So in some sense 1,000,000! is a functional infinite integer. I am probably 10-15 videos in here and have seen several of your appearances on other math podcasts, and I very much like the premise of your methods. It seems much more rational, and it seems that it is pretty much the method used when we need to apply the math to a physical process. Most of my problems with the continuum are around ideas like the set of rational numbers is countable set. Like somehow you can complete this set and say it’s a countable infinity. But if this concept makes sense, how can there be any irrational numbers at all, if you have written down all of the rational numbers over an interval shouldn’t there be no space left on the number line to fill in? My other big struggle with things like root 2 are if we make a right triangle with 2 sides length 1 such that the shape is inclosed. The perimeter around the triangle would be 1+1+root 2. But is this the inner perimeter or outer perimeter? We cannot draw visible lines of width zero. Any shape that we draw, will not have an equal inner and outer perimeter functionally. If we could have lines of width zero maybe it would be possible to have root 2.

[@carly09et]

this was what drove Cantor "mad" - this glitched me in the 80's - Godel proved it cannot be fixed! Just finer and finer finite approxamations. The area of all circles is zero ... The area of a disk inclosed by a circle???

[@rajendralekhwar4131]

Excellent professor..!

[@dehnsurgeon]

as an aside, haskell has in-built libraries to work with computable numbers in exact form

[@david-hogarty]

Tell me more... links?

[@dehnsurgeon]

@@david-hogarty look up "Exact Real Arithmetic in Haskell"

[@oker59]

you're telling me there's a last integer(or natural number). What's the last integer if there's no infinity?

[@oker59]

is google the last integer?

[@Omeomeom]

I think it's more valuable to figure a really good alternative defintion of the real numbers that isn't decimal expansion. The definition shoulbe be something that that's realyl easy to write computer programs with. No huge library or anything just a really primitive object.

[@hellNo116]

I have a conjecture as a computer scientist. The anxiety of what stated here it is not lost on young mathematicians nor computer scientists. We seem to joke about those things all the time. As a collective we know the contradiction is there. And we make joke and laugh about it. However I don't feel like the majority of us have the vocabulary to talk about it or don't care enough about it or we simply didn't hit that inevitable wall like matheticians did 100 years ago so we simply decided to ignore it. I am very intrigued and I would happily sit down and shut up and here a course worth of videos before my next comment if I said something profoundly stupid, since this is a matter that seems fascinating to me

[@aaronmartens2903]

Has anyone concisely explained how to generate a rule (formula) to describe the digits of an arbitrary real decimal? Well how can we point to something that doesn't exist much less describe it. The next question might be can we create a collection of rules that cover the real numbers? I have a feeling any collection of rules would simply mirror these arbitrary real decimals (for instance dedekind cuts need to specify an infinite sequence of rationals that point to a real number). Although i appreciate the idea being brought to me that real numbers are tenuous, without a strong counter argument the notion becomes tired.

[@danaferguson3342]

thank you.. your talk today reminds me of Harold Aspden's writings. He has been one of the foundations for my latest deep dive into over unity design for generators, motors and the aether connection .... somehow KZbin got the algorithm right for me to land on your site. I still like (and use) my analog computers for solving those pesky D E's that are insoluble to Mathematics. Get's rid of the pesky gaps in computer approximations. :) ✨

[@hihoktf]

I recently put some thoughts down on paper where I asked myself the question "What is mathematics most fundamentally?" My best answer was "A difference engine, and not specifically Babbage's 1820's machine, but as a general idea. Mathematics is a tool to differentiate between two different things, and by how much they are different; even to determine if they are different at all in the first place." For my taste, in that infinity is unbounded and randomly open to the universe, the use of infinity in mathematics violates its most fundamental use as a difference engine. Can mathematics in its common present formulation tell me if '3.14...' and '3.14159...' are different? How do you compare '...'s?

[@rgerk]

They are just approximations of Pi. Like e^((sqrt(163) π)/3) isn't equal 640320 but if you put this into a calculator, it will give you this number.

[@hihoktf]

@@rgerk Why did you think they were approximations for pi? I never said that. The fact you might think they are is my point. Pure mathematics can't assume that. There is no legitimate reason for anyone to assume the two "numbers" I gave are one and the same thing, nor that either of them is intended to be a partial representation of pi. It's this assumption that is the problem because the assumption is conflated with logical reasoning. My argument is that it's a fundamental flaw in the current formulation of math theory built on set theory with infinity included.

[@rgerk]

@@hihoktf Because they are. The second is more precise. You can't write irrational numbers in decimal form. Three dots doesn't mean anything, only that you didn't finish to write the number.

[@hihoktf]

@@rgerk You are the one that brought up pi. I didn't. How do you know my next three digits weren't '234' in both cases? My point is that you made an assumption, and that your assumption does not constitute a logical proof.

[@rgerk]

@@hihoktf 3.14234 is still an approximation for Pi better then 3.14, but worse than 3.14159. If you are approximating "e" then 3.14 is better. All of this in base 10. It's better to use fractions, series, continued fractions and its convergents.

[@LemoUtan]

If one's concerned about the actual existence of real numbers in the universe then won't it be necessary to demonstrate that dimensionless quantities in the universe - such as (for example) the fine structure constant - are either (a) rational or (b) not constant? Either of which would be a highly interesting insight into 'reality'. Constant-rational would, presumably, involve demonstrating certain comibatorics of our universe, whereas if not constant then its value's evolution would have to, in some way, 'snap through' rationals along its path - which would (presumably) imply quantisation.

[@scavengerethic]

Another two possibilities are: (c) the fine structure constant is a non-rational but finitely-specifiable number. If you include them but exclude uncomputable numbers you still have a fairly sane number system. Or (d) the fine structure constant isn't a number, because we shouldn't confuse our mathematical models of reality with reality itself.

[@LemoUtan]

@@scavengerethic By computable, but non-rational, you mean something like algebraic? So you'd 'do' arithmetic in physics with a field extension (much like it does with with √-1 I suppose)? Not being a number at all is a bit tricky since this particular 'thing' is a ratio of measurable things - but admittedly only if, as you say, we don't confuse e.g. 'the speed of light' (i.e. to a photon, 'speed' is meaningless) or 'permittivity of spacetime' etc with something actual.

[@jesperandersson889]

good ideas!!!

[@akswrkzvyuu7jhd]

At this point a Social Media warning notice is needed. The Axiom of Choice only works in a Buffet Restaurant.

[@GiorgiTsutskiridze1990]

Thank you for this amazing video. I am not a mathematician but I am very interested in deeply understanding the decimal arithmetic that we currently uniformly accept in schools and use and the flaws it has as you say in the video. The video was very insightful and motivating and looking forward to the next one where as you promise you will explain the problems of decimal arithmetic. BTW, I meter questioned the foundations of decimal numbers and their arithmetic and it looked natural when I was taught at school but now I became interested in its foundations? Would you recommend any material where this stuff is explained, I mean all the foundations of decimal arithmetic and why it works the way it works in our everyday life?

[@dehnsurgeon]

I appreciate you viewpoint, but I think its important to note that computers can (in principle) do all the mathematics that mathematicians want to do.

[@karenthebroker5618]

Are you saying this because the continuous/infinite ideas in math are built off finite logical operations and satisfaction criterion?

[@dehnsurgeon]

@@karenthebroker5618 Not exactly but kind-of. Type theories are more suitable than set theory since the Curry-Howard correspondence is so powerful.

[@karenthebroker5618]

@@dehnsurgeon It seemed like you were saying any mathematical operation a human mind could do, a computer could as well(in principle).

[@dehnsurgeon]

@@karenthebroker5618 that's probably not true, but anything constructive can be encoded by computation, and nearly everything mathematicians want to do can be done constructively

[@karenthebroker5618]

This doesn't seem to be true in the modern framework. I was under the impression that the majority of mathematicians are concerned with existence and non-constructive proofs. Construction is reserved for approximate/applied math.

[@juditcsohany5]

Automatic english subtitle, please?

[@value21value]

The issue has a lot more to do with this thing called identity defense in humans than anyhting else.It all comes down to why would so many "smart" people actually BELIEVE in such things?Once you insert that word into a discussion where objectivity must be tha ground 0 problems and misconceptions are inevitable.

[@strangeWaters]

There are some aspects of "infinite objects" that I do think can be grounded finitely -- there's various notions of "coalgebra", terms that "loop infinitely", which get explained in a nonsensical way, but I've actually come across them while programming so I understand them a little better. If you have a data structure struct NumberTerm { digit: 0..9, next: Reference } And you arrange a collection of these objects so that their "next" pointers form a cycle, you have a finite model of an "infinite" object. Of course any calculations you do to the "infinite" object amount to doing calculations on that *finite* graph. This actually does come up in practice, e.g. when doing range analysis for variables in a compiler; these algorithms often have an "all at once" flavor -- if you have a more complicated reference loop structure than a simple cycle you might apply "bisimulation" to collapse redundant nodes, which involves starting at the coarsest spot in the (finite!) partition lattice on your collection of nodes and making cuts until you arrive at some fixed point. You could also add nodes to a set until you reach a fixed point, e.g. for "taint analysis" in computer security. (For a great introduction to these sorts of algorithm, see the paper Introducing Fixed-Point Iteration Early in a Compiler Course by Max Hailperin). So for me an "infinite decimal" is just a pointed labeled cycle, which is a perfectly respectable finite combinatorial object, and more generally "infinite trees" and so on can be seen as "databases of nodes" containing cross-reference loops. Of course labeling all these as "infinite" is actually completely wrong! There's a habit of treeifying graphs to understand them, but if you try to treeify these graphs you will never stop -- but despite this one can still do math with them. I think this leads people to the idea of "infinite things, but they still make sense"; generally one of the terms involved in such a situation eventually references itself again, but traditional math doesn't have a notion of terms that "loop", only Platonic Ideals Existing somewhere, so you end up with people trying to explain them as "infinite objects that really exist no I can't show you trust me". Imo this falls apart for transcendentals (but you can prove they "exist" in a category-theoretic framework for any "coalgebra functor" aka node data structure, iirc Sangiorgi does this in his Introduction to Bisimulation and Coinduction textbook. Of course since the whole book is described in terms of "infinities", it's harder to spot the slight of hand when you switch from cyclic graphs to transcendentals.)

[@ChristAliveForevermore]

You seperate "traditional math" from computer science as if one didn't give rise to the other. Why should mathematics not be based on Platonic Ideals if Gödel's Incompleteness Theorem *proved* that mathematics is fundamentally non-logical (unlike computer science which runs on Boolean Algebra)? The philosophical conclusions that metamathematicians have come to are not at all arbitrary.

[@dehnsurgeon]

it doesn't matter that the objects are in some sense infinite, we are ok as long as we can communicate our ideas in a finite amount of time and space

[@peterjansen7929]

This is a relevant observation, particularly as there are hardly any 'numbers' in Cantor's second diagonal 'proof' that can be communicated in a finite amount of time and space! √2, e, π, √π, these are all finite symbols that can actually be presented to us. But Cantor would let us have 'infinitely' many 'numbers' 'defined' by 'infinitely' long non-recurring sequences of digits, that is 'infinitely' large 'symbols' for 'ideas' that nobody ever had or ever could have. Then he would have us draw a conclusion from a supposed contradiction between these non-notions!

[@dehnsurgeon]

@@peterjansen7929 at lot of people in the comment section would greatly benefit from taking the time to educate themselves about mathematics. you can define those constants in finite time and space, but the mathematical tools you will need take effort to build and understand - I suggest you put that effort in

[@dehnsurgeon]

@@peterjansen7929 btw, all those constants are computable, so they can easily be represented in a computer

[@TheDummbob]

@@dehnsurgeon But there are noncomputable numbers in the reals! And they can never be communicated nor written down nor specified nor defined in a finite amount of time!

[@dehnsurgeon]

@@TheDummbob technically, some specific uncomputable numbers can be communicated in a finite amount of time (such as Chaitin's constant) and more generally, the concept of real numbers can be communicated in a finite amount of time

[@yahavitah2791]

It's fair to criticize the construction of the reals but i dont see why its impossible to have a construction (in fact i have a theory for reals and more stuff and i may upload some vids sometime)

[@tinaryan4023]

Orthodoxy loathes a Paradigm Shift

[@dansanger5340]

A mathematics without pi and e (or even root 2) seems impoverished.

[@ThePallidor]

Pi, e, and root 2 are (sets of) processes, not numbers. They generate successive numbers that can be used in real-world calculation. The dream of "pure math" has to be let go.

[@dkenworthy2036]

10/9 represents infinity put this is a computer and the calculation will never stop. 10/9 takes half a second to type in. I think i get what you are saying

[@thesmallestatom]

Axiom of Choice is the Oracle of Delphi.

[@TheDummbob]

Nice! I'm looking forward to finally delve into the real problem by looking at the (nonexisting?) arithmetic of the reals, especially I would like to see how the problem presents itself when presented with "uncalculable" real numbers. Until now I'm not totally convinced that the reals are bogus, though I found your videos very refreshing in talking about this subject and all its subtleties, I definately learned alot of new stuff and you got me interested in the real numbers and the "continuum"! Especially as a physicist I like the idea that the concept of the continuum may have been "translated" in a somehow faulty manner from our (seemingly) direct experience into the realm of logic and mathematics, which of course would be fatal for building physical theories ontop of it! And when considering that the physics question of the millenium, namely how to marry GR and QM, is exactly about how things are at the very smallest distances, then it gets really interesting! :D

[@whig01]

I can represent any algebraic number by a matrix, and so they are concrete numbers. Exponentiation requires infinite matrix expansion, so these cannot be considered concrete in that form.

[@20-sideddice13]

I do not understand why you dislike so much the axiom of choice. This axiom does not imply the existence of infinite sets. We don't even need the axiom of choice to build the reals. Why don't you attack the axiom of infinity ? Wouldn't it be more appropriate ?

[@johnk8392]

It's simple, prof. Wilderberg is clear about this: the axiom of choice leads to infinite information, infinite entropy and infinite randomness yet fixed, which is nowhere to be found on the real world. The axiom states, that there are infinite sequence of numbers. With each value in the sequence has ano relation to eachother at all, which necessarily requires that something, somehow has chosen all these numbers ad infinitum and they are forever fixed. That is simply totally absurd and nonsensical.

[@20-sideddice13]

@@johnk8392 just go look up the axiom of choice on Wikipedia. There is no question of infinity. It states that the cartesian product of non empty sets is non empty. Nothing more. Of course the interest of the axiom of choice is in the case of infinite cartesian products, but if you don't have infinite sets then the axiom is trivially true.

[@johnk8392]

@@20-sideddice13 I'm just stating what wilderberg says about that axiom in his videos. Somehow it is connected to the fact that the infinite decimal expansion is completely random, as each next digit is being selected by "choice", implying that there is no way to predict its value. This is not my field, so I cannot comment further. It seems that you are much more knowledgeable in this. Maybe you understand what the prof. talks about this.

[@20-sideddice13]

@@johnk8392 sorry to have been this unpleasant. I should not have written like this. This position of the author of the video angers me for the reasons i wrote above. Have a nice day.

[@johnk8392]

@@20-sideddice13 not at all unpleasant. In fact you piqued my interest and made me critically think about it in more detail. I follow Wilderberg, because I think he does makes sense to me (about the various issues with the Reals, infinity etc) because I am a computer engineer and I have to deal with these kinds of issues. I thought he was on to something. I didn't know how other mathematicians see his views. You are probably one of them, and this is interesting to me. I would love to hear more criticism on his view to be frank. Anyway, have a nice day too!

[@岩男沢山]

If I can define a world that is boundless in its definition (the generative rules) but has within those rules a way to limit evaluation to the examined cases, then I can make it real in a computer. It is a lazy universe. It isn't clear to me that our own universe isn't this type of place. The "infinite specification" version is simply not something that can be evaluated, examined, concieved in a complete way, or shown to exist at all. If any layer of our world is "infinite" in its definition then it seems it must be, at best, lazily defined.

[@dehnsurgeon]

Can you give your opinion on Homotopy Type Theory (HoTT)?

[@njwildberger]

@Thomas King I have not studied it well enough to say. But the once or twice I looked at it, I felt that the basic definitions were not clear enough for me to quickly grasp them. Perhaps that was my fault for not applying myself. But I like to have examples along the way that illustrate the "abstract concepts" that are being introduced, and I felt their absence.

[@dehnsurgeon]

@@njwildberger Knowing how most type theories work should be enough to start learning HoTT. Also, you can probably strip away the "abstractness" by using computer implementations, found in proof assistants.

[@dehnsurgeon]

@@njwildberger Have you ever tried learning a functional programming language like Haskell? If you learn Haskell, you can then start to look at languages like Agda, Coq and ldris. The basic starting point for all type theories is simply typed lambda calculus.

[@njwildberger]

@@dehnsurgeon Not being versed in computer science sadly, I am not familiar with type theories. And unfortunately I do not trust proof assistants. In any case I want to stick with pure mathematics here.

[@dehnsurgeon]

@@njwildberger You can prove proof assistants correct, so there's nothing to worry about. Also, I was only suggesting it as means to get hands on with type theoretic ideas. This kind of stuff is pure mathematics and is one of the many logical foundations to choose from. From what I know about your opinions on the foundations of pure mathematics, I get the impression that a more computational approach would be appreciated.

[@jessewolf6806]

Any pure mathematician who believes that mathematics must be expressible in terms of a computer program is not “pure”. There is a reason that none of the opinions expressed in this video, as far as I know, are discussed in ANY top tier graduate school of mathematics or peer reviewed mathematics journal. I am not an expert in the logical foundations of mathematics - but I hold a PhD in math - and as far as I can tell the Professor’s viewpoint is so far out of the mainstream as to be not credible. If he is in fact actually correct that the real numbers are fictitious he must present a much more rigorous argument.

[@_ARCATEC_]

✍️🤓

[@rickshafer6688]

The unreal number system. More at pogosticks.

[@rickshafer6688]

Objects are as close a definition as we'll get. All indeterminate numbers are objects then. Seems lacking a God thing.

[@christopherellis2663]

None, one, some... 0, 1, x.

[@bernardoxbm]

The continuum is dead

[@ebanfield]

Not dead so long as W is alive.