Відео

I am learning computer architecture and this whole playlist is incredibly illuminating!

I would be happier if you were more careful about not conflating the ideas of truth and provability.

It would seem the idea of a number being large or small is completely relative. If you measure the distance to work in light years you will get a very small number, where even small changes in the distance will represent large changes in location. Or you could measure the distance to work in picometers, now your distances will be very large, but even large differences in the number of picometers will represent a small change physically. In reality we tend to use a unit of measurement that balances the quantity of units needed with the accuracy which we want to have in the measurement. But there is no unique unit. We can make a measurement very large or very small by changing our scale of the unit. But there are many issues making physical measurements with units that are not scaled properly for what we are trying to measure. It gets inaccurate quickly when using measuring device that is calibrated for a different scale of measurement.

Hi, on 28:17, it writes S(A_1, A_2) = sin^2(\theta(A_1, A_2)). Should it be sinh instead of sin? Otherwise, how can we get infinitely large numbers?

One of my biggest issues with real numbers is where exactly are they located? It seems we can only define their position accurately to a finite number of digits, which we can then express as a rational number. And when calculating its expansion it can be listed as a rational number at every step or every interval. But since it is an infinite process, the exact calculation can never be completed. However if the process could be completed would the final number not be a rational number as well? It’s a rational number at each step, thus intuitively it seems that it should be a rational number at a final step. But it seems that irrational numbers for the most part are not used in computation. Instead we use a rational approximation using n digits of the expansion to have some specific level of precision or accuracy in a final calculation. We can give it a new symbol on paper, but when we need to translate that to a physical construction it must be approximated. My second issue is with the idea that numbers can extend infinitely far to the right of a decimal but not infinitely far to the left of a decimal. Like what is the number …11111111? Or the series of the powers of 10? So I have found these videos very interesting. To me the first geometric sequence 1, 1/2, 1/4, 1/8, 1/16, 1/32 is not heading to zero, but to an infinitesimal. Where if you take the last number in the sequence and multiply it by 2 an infinite number of times it will give you an answer of 1. It might get arbitrarily close to zero, but it will never get to zero. And this value that it is approaching is a distinct infinitesimal. Because if you take this infinitesimal and multiply everything by 3 instead of 2, you should not get back to 1 after the infinite amount of steps. But I think this really gets into what does it mean to be continuous? If space is continuous then you have some of the infinity paradox’s like Xeno’s paradoxes.

Thankyou

Thankyou

Thank you

Good lesson thanks

Good lesson

i know what hes trying to communicate but its just horrendously stupid, hes basically trying to say something on the lines of "analysts believe that the infinite series that was shown can be determined to be 1, theroefore they must also believe that the twin prime conjecture is true".

tell me you dont understand set theory without telling me you dont understand set theory...

360 ×360

I always had a problem with pretending zero is a counting number, I could/can only accept it is a placeholder in orders of magnitude. Zero means no number and I think a lot of problemc in arithmetics are solved by acknowledging zero is not a number. Division by zero means nothing 3/0 is similar to 3/happy, does not have any obvious meaning. 3*0 also is different, maybe it means you have 3 nothings? I understand it is convinient to use zero like a number but I believe it can fool us in the long run if we forget 0 is not a number like others. So we bend the rules to include zero as a counting number but why must the noncounting number 0 be a counting number like the counting ones? I think zero is its own class and maybe we could use it differently? Maybe math is the laws of physics and zero does not exist per definition? I would appreciate if someone could show me why zero is carelessly thrown around like it had a counting meaning. Zero and infinity are special cases that causes problems, impossible magnitudes.

This is so stupid. Can't believe that you are teaching mathematics. There is absolutely no problem in the definition of the real numbers as equivalence classes of cauchy-sequences... You can argue that you are not accepting ZF... but that's another story...

If we don’t have the correct language to even state what’s actually going on when we talk about the Riemann Hypothesis what are some ideas you have about how we might explore the deep and beautiful phenomena underlying it using strictly finite mathematics?

Hi, I came here to leave a comment after watching your video. Your video is absolutely amazing, and I really enjoy your teaching style. It's friendly and approachable, and I'm especially impressed by your steady pace of speech, without any awkward filler words! I know many young students like to rush into some big theorems when beginning a new subject, only those with certain learning experiences will understand how important the inspiration from the history of mathematics and examples is.

So here what I noticed about patterns : It is always divisible by 3 because 10^x+23 = 1+2+3=6 which is divisible by 3 if you divide (10^x+23)/3 you will get 3*(x-2)41 where 3*(x-2) is the number of repetition of 3 , for example ((10^5)+23)/3= 33341 with that expression is 3*(5-2)41=33341 11 factor repeated in odd powers, 41 repeated in 5k+2 13 repeated in 6k+4

Your phrase is not a natural number. It’s just a phrase that sounds like it describes a natural number

Professor Wildberger reminds me of Captain Janeway 's doctor on Star trek. He will be teaching forever 😊❤

For a so-called real number to be called a 'computable number', all that is required is that we can write an algorithm that would determine its n-th digit. My main reservation about terms like "computable number" and "computable function" is that they don't match the common usage of "computable", as such, they appear to have slippery definitions. The common understanding of "computable", as I see it, is "capable of being computed". This implies a computation process leading to a definite and precise answer. The mathematical definitions are crafted with the clear intention to categorise real numbers like √2 as being 'computable'. But to say that √2 is computable suggests that we can compute √2 to infinite precision, which is absurd. Based on the common understanding of "computable", it's evident that √2 isn't computable. Much of mathematics, particularly concerning real numbers, seems rife with misleading expressions. I believe that if we employ a term like "computable", it should not be defined in a manner that suggests the computation of infinite values is feasible. It is frustrating to witness mathematics perpetuate such practices, which only serve to reinforce the mistaken belief that infinite processes can be completed. There is a more in-depth discussion about these matters at mathforums dot com slash computer-science. Look for my comments in the thread called: About why I believe that the "What Computers Can't Do" argument (i.e. the halting problem proof as applied to real world computers) is not valid.

Your system looks really great for scales, but is very confusing for chords. You get multiple different black/white patterns depending on whether the chord root is white or black. Also, the major and minor chord patterns look too similar.

How the Basel problem could be revised ? What’s about the solution ? Should we consider a solution does not exist or that the solution is « unreachable/un computable » ? ua-cam.com/video/JAr512hLsEU/v-deo.htmlsi=RKN1JVOCvzMc3jQe , we could probably revisit the definition of the equal symbol and of an equality to be more rigorous to make the distinction between an equality one can evaluate and something approaching « a point » at the infinity that is unreachable

You can puff and wheeze all you like, but real numbers fall out of even the most basic geometric reasoning, regardless of the limitations of decimal representation. You are complaining that 'real numbers can't be defined' in that their definition is unsatisfying, but I can easily claim the same for other foundational definitions, like natural numbers, which are typically defined in terms of the metaphysically questionable object known as a set. Numbers have no requirement to appeal to your limited phenomeonological intuitions about time and space. They're there right in front of us and we can soundly reason about them. You are picking a choosing what you accept as sound but it's completely arbitrary, you can't even demonstrate negative numbers 'exist' in your schema. Meanwhile I can make a sound geometric demonstration of sqrt(2) using a square. You are deeply confused about the big picture, and if you had a shred of intellectual honesty, you would realise that following your logic would make all mathematics impossible.

but the process of "creating" also applies to the series 1/2^n or not? it's just that we can specify an exact number 2 as the limit which ist 1.99999 .... infinite 9s. In both cases that limit will never be reached but arbitrarily gotten close to.

Is this inspired by grassmann and Clifford?I am reading, common sense in the exact sciences by Clifford now, this way of thinking with m-sets seems to be the foundation for understanding geometric algebra.

What is your opinion on formalism? We define "real numbers" and "infinite sets" formally without actually believing in them, it all reduces to finite string manipulation. In this view, when someone says something like "there is an irrational number", they don't actually mean there exists such thing, it is just a shorthand for a certain formula being provable in a certain formal system, and formulas are finite and proofs are finite too.

I've got q question; How can the reals be denser in irrational numbers if every irrational number has an arbitrarily close approximation in rational numbers? It is not true that the irrational have infinitely many representatives between any two rational numbers because between any two irrational numbers is an irrational number. I understand that the converse is also true. That the rational numbers can be approximated arbitrarily closely by irrational numbers, theoretically., and that between any two irrational numbers is a rational number. What is the difference in symmetry that could account for the density results? I realize that we have the concept of countability, but is there any other explanation, for example symmetry of some kind, that would justify the results?

At around 17 minutes, C(Q) is an infinite dimensional vector space. How does it convey the idea of A^2. Can you please help me to understand it?

Thanks, very illuminating. A further question. In the third case we can we can represent the decimal string as a continued fraction reading of a list of rational convergents. Can we thus define distinction between 2nd and 3rd case as continued fraction with a single final convergent vs. open ended reading of many convergents? Are the other complexities involved that make this definition incomplete?

Thank you for great videos Mr. @njwildberger, what about sequences that converge slower than 1/n ? Say, can we talk about convergence in a sense that -k/n <= (p(n) - A)^2 <= k/n ?

Finally stumbled upon this video that's already 3 years old, so my comment is a bit late. I have always rejected the B-T paradox. It is not a paradox. It yields an impossibility, i.e. a contradiction, thus falsifying at least one of the premises. Ex falso sequitur quodlibet, from falsehood follows anything you like. My simple reasoning is that pretending to touch infinity and then come back home invalidates ANY proof. Infinity is uncome-at-able and NOT a number. It easily beats Graham's number plus one... It is like "proving" the echo of an infinitely deep well, not by shouting and waiting until you hear it, but by mere reasoning. A bit like those morons saying infinity minus infinity equals pi. Yes, pi plus infinity is infinite, but you cannot simply move infinity to the other side of the equals sign and swap its sign. Albert Einstein: Propositions obtained by pure logic are completely empty with regard to reality. And yes, he DID say this (at the Herbert Spencer lecture, Oxford, June 10, 1933).

No matter where I roam, I will return to the University of New South Wales

Great explanation of homogenous polynomials

So, the properties of the rule that generates the series are mistakenly forced into the category of numbers or number-like mental objects. In fact, a certain algorithm-analysis (based on arithmetics) is being dressed up as "real" analysis. Thank you, professor Wildberger. I get your point, I am woken up. I hope that some of your colleagues will join you soon in reconfiguring mathematics. You have a lot of work to do.

Of course there are ways to define real numbers rigorously. The only way to make the claim that "real numbers do not exist" is to use a different notion of existence to the rest of the mathematical community. This may legitimate, but I'm sad that Wildberger is not upfront about this. It is not clear to me that Wildberger's notion of existence is philosophically consistent, but that may just be a lack of understanding on my part.

Good stuff Norman. I'm listening to this in June 2024 and think that if you held this discussion with Anthropic's Claude you would be even more delighted!

"real" numbers are simply representative of unbounded computations. indeed they're quite lazily defined as they allow for not only non-computable "numbers", but un-nameable ones. gregory chaitin is a closet skeptic of the idea of real numbers too, despite having studied them from philosophy/foundations of mathematics. personally they seem like a lazy way to avoid having to better model more complex algebraic objects by just throwing every possible unbounded stream of digits under the same name

I would even go further than say Galileo's physical equations predict Einstein's equations; I would argue Einstein's physical equations can be directly derived from them, because they are in fact the same: Galileo's equations that define the physics of sound propagation and Einstein's equations which define the physics of light are similar. Add to this, the fact electromagnetic radiation which have shorter wavelengths is sensed as light and radiation with longer wavelengths is sensed as temperature... I ask you: Is it maybe even possible that because there are certain wavelengths of radiation which effect living cells, which a living organism then attempts to repair as an adaptation -- perhaps there existed an era upon a relevant time scale, with an abundance of this short wavelength radiation which perpetuated the evolution of all organic cells, until the radiation somehow vanished or extended in wavelength to non-reactive energy bands ?​​​​​​​​​​​​​​​​ 1. Physics similarities: It is correct that there are similarities between the equations describing sound propagation and those describing light. Both are wave phenomena, and many wave equations share similar mathematical structures. 2. Electromagnetic spectrum: The observation about different parts of the electromagnetic spectrum being sensed as light or heat is also correct. This is due to how our bodies have evolved to detect and interpret different wavelengths of electromagnetic radiation. 3. Cellular effects of radiation: It's true that certain wavelengths of radiation can affect living cells. This ranges from beneficial effects (like vitamin D production from UV light) to harmful ones (like DNA damage from high-energy radiation). 4. Evolutionary hypothesis: The suggestion about a potential era of abundant short-wavelength radiation influencing cellular evolution: While speculative, it's not entirely out of the realm of possibility. Here are four more points to consider: a) Early Earth conditions: The early Earth had different atmospheric composition and potentially higher levels of certain types of radiation reaching the surface. b) UV radiation and early life: Some theories suggest that UV radiation played a role in the formation of complex organic molecules that led to the origin of life. c) Cellular repair mechanisms: Many organisms have developed DNA repair mechanisms, which could have evolved in response to radiation exposure! d) Changing conditions: Over geological time, Earth's atmosphere and radiation environment have changed significantly. However, here are the challenges to my hypothesis: 1. Timescales: The evolution of complex cellular repair mechanisms typically occurs over very long periods, while changes in radiation levels might happen more rapidly. 2. Persistence of repair mechanisms: If the radiation vanished or became non-reactive, we might expect these repair mechanisms to be lost over time if they were no longer beneficial. 3. Lack of evidence: Currently, we don't have direct evidence of such a specific radiation-rich era coinciding with critical stages of cellular evolution. To further highlight the important interplay between environmental conditions and biological evolution, which proves my hypothesis, would require further research and evidence in the fields of astrobiology and evolutionary biology.

Thanks, Prof. W., for this interesting video. I remember when I first saw the claim made in a class that real numbers could be defined as equivalence classes of Cauchy sequences, followed of course by a hand wave over the issue of proving that those things comprise a field. As I understand it, that briefly is your objection to the real numbers - even if you accepted that those equivalence classes exist, would they provably form a field? I wonder if it would work to allow that yes, a Cauchy sequence of rationals has a limit, but almost always that limit is just a geometric point but not a number. Then pi would be something, a geometric object, but for engineering purposes we have to use partial sums of a series whose nonnumeric “sum” is pi.

On a serious note...The more I listen to you about real numbers, the more it reminds me of Wolfram's "Computational Irreducibility", with the added proviso of requiring infinite computing power. Computational Irreducibility can occur in finite steps. I don't know if that is pertinent or not. Are finite and non-finite computationally irreducible systems equivalent? This also reminds me of your analysis of whole numbers and their fractal nature. There is something going on here with numbers that has not been clearly specified yet; some big insight to take things to a whole new understanding. I would guess that computational irreducibility will be an essential ingredient in whatever this next notion of numbers turns out to be. Thanks for all you do!

"...in to another universe where there's an infinite amount of room" LOL!

Imagine if this guy had pursued something worthwhile with his life...

Ok, thanks for a good analysis on limits. I think lots of things could be added though. The reason for real numbers in the first place must be to be able to successfully deal with continuum problems. You simply add "all" irrational points on the line and call them real. Rational numbers have the advantage of being point specific, as well as whole numbers, but for completeness reasons, that's not true of the reals. There is also the issue of mixing irrational numbers (irrational reals) with rational numbers or rational real numbers. This is not trivial, since the results of calculations will differ depending on which model you pick. Take the rational number 1/3 for example; is it really exactly equal in every way to 0.333...? Certainly not if you define 0.333... to be a real instead of a rational number. A real number has a sort of "fuzziness" to it, where values (or more specifically, points on a geomertical line) that are infinitesimally close count as the same real number, something that doesn't hold for rationals, since real and rational number objects have very different definitions. So, who's to blame for the confusion about these concepts?

What do you think of using infinitesimals as in nonstandard analysis to define calculus, instead of the epsilon-delta limit approach?

1+1 cant equal 2 bc there are infinite numbers between 1 and 2

Of the scenarios you've provided, whereas the first two posites "Does the series converge, and if so, onto what real numbered value ?", I view the third example series as an approximation problem, up to an arbitrary precision, unrelated to the two. However the third example has monstrous ramifications in the area of artificial neural networks verification, which largely base their operational characteristics on mathematical models! The third example series you've provided is an excellent example of the inherent problem involved in verifying most artificial neural network models, and also why most of the verification methods of these systems embedded in our automotive, healthcare, food industry, and military systems, even up to a 99.99% reliability, is wildly insufficient.

I don't really think that not being able to fully describe a number is a problem. There are just more numbers than ways to describe or compute them, but we can still find properties of numbers, even ones we can't define. The problem doesn't get much better when you move onto things like functions.

Hey, I really appreciate the video! Now I know why you didn't respond further to our little discussion on one of your previous videos :) So let me try to push back a little, again. I'm trying to learn here ;) I suppose you don't accept the notion that a "real" (say, Platonic) circle exists, as then I could simply rhetorically say to you: "how is the ratio of its circumference to its diameter not 'real'," or even more fundamentally, "how is its circumference or its diameter not a 'real' distance, if the circle really exists." You know, thinking of it right now, I wonder whether a real number might not simply exist in any property of the world that can't be described as a rational number from within a certain unit-system, if you know what I mean. People who think space itself is discrete would probably disagree though, but that hasn't been proven, and strictly speaking it wouldn't imply that 'distance' is fundamentally rational. But maybe we don't need to posit that a perfect circle exists somewhere in abstract whatever. What's wrong with defining an irrational 'number' as a limit of numbers? I think what I argued for in my previous comments, and what you expressed your acceptance of in this video, might still hold, albeit in a more nuanced way. We don't have to say that the series sums to something that physically exists in this cosmos, but we can mathematically prove that the sum moves less and less around as you take more and more terms, so much so in fact that it kind of falls deeper and deeper into a narrowing pit and never comes out. So real numbers are like the black holes of the number line, things that fall in might never reach the center, but they never (sorta) go away from it anymore either. Why not give this center a name and try to reckon with its influence, beyond reach though it may be? Furthermore, there are rational numbers that aren't containable in this universe either. Would you deny their existence too? I'm really curious about what you would reply to this.

I am no sociologist but it looks like this is related to finite semiotics.