Unpredictability, Undecidability, and Uncomputability

“As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” – Albert Einstein

I think it is also important to state clearly: This is a BEAUTIFUL DRESS, and you look marvelous on this video.

A good and important video, although I not quite agree with everything said. Around 90% though...

Nice dress.
I know it is not physics, but i really like it.

Computer Scientist here. The thing with undecidability is that computer scientists work with problems which they abstract into languages. A language is basically a set of strings if you like, which are a subset of a concatinating powerset of an alphabet of different symbols. This sounds crazy, but hear me out. You surely know the diagonalization argument for real numbers that if you take the first digit of the first number (row) and add some constant (besides zero) to differ it from the line we took it from and then continue with the second number of the second line and so on, you will get a whole new number out of it that hasn't been counted yet. So we have an uncountable set. But the number of possible turing machines (or other computational models) is countable infinite. But the Set of languages are uncountable, which has been proven using a similar proof like the diagonalization argument of cantor. But if the turing machines (from which each one accepts only 1 language) define what is computable, then there must be a set of problems which are undecidable. Quick side node: decidability defines the following: If a turing machine can tell in an finite number of steps if a word (a substring) is part of a language or not. But you can also end up in infinite loops without ever getting out... I hope i could clear that a little bit up for you (commenters)

Bad question. Nothing is unpredictable, undecidable, or uncomputable. The question is their accuracy, which depends on sufficient accurate data. So the answer is whether there is sufficient accurate date being used to make accurate predictions, decisions, and computations.

"Nothing real is infinite." What about real numbers?😏😏

I couln't understand anything meaningful from what Dr. Sabine said on this video. Perhaps others are much smarter than me? The image of a butterfly is not an explanation for the "butterfly effect."

1;45 Should I take it that light isn't infinite or nothing real :) Keep me away from those guys.

The mathematical truths expressed in the incompleteness theorems are nothing but the engine of nature. Stephen Hawking once asked what breathes fire into the equations. I think I have the answer. The incompleteness of mathematics, expressed in the incompleteness of the surrounding world.

I think these three things are closely related to Heisenberg's uncertainty principle. At least one of these three things is related to Heisenberg's uncertainty principle. Heisenberg's uncertainty principle is a fundamental fact about the matter in our universe. We have one planks constant of unpredictability unmeasurability on every ability we got it. Consider a particle called a railroad train. It is represented by a wavelength just like everything else a debrogoli wave. The uncertainty in the position of something in massive as a railroad train is very small.

She is great, could she just put on a Startreck costume, im serious it would work.

the 42 made me smile

I think that the situation with computing real numbers is slightly more complicated:
-in physics there are irrational numbers like pi oraz some square roots. Indeed, when we calculate something we calculate it to a given precision but these constants are present in the theory so I woulnd say that irrational numbers are not present in physics. However:
-there is a concept of computable real numbers: practically all of numbers which we encounter in everyday life (many of them irrational) are computable. For these numbers there is an algorithm to compute them with arbitrary precision
-there are numbers which are definable, i.e. can be defined within the language of mathematics. This set is larger than the set of computable numbers i.e. there are numbers which can be defined but for which there is no algorithm to compute it with arbitrary precision. Chaitin's constant is an example of such number. If this constant would be computable then the halting problem would be decidable which is not the case. Another great source of examples are so very largeso called Busy Beaver numbers
-but still even the set of definable numbers is countable so those numbers do not exhaust all fo real numbers. In this sense most of real numbers are even undefinable-but there are there to form the SET of real numbers which has decent properties like completness etc. even if we cannot pin down most of these numbers.

I object to you considering mathematics as opposed to science, for example when you say “this is an interesting mathematical curiosity but has no interest for science”. Such statements could mislead people with small knowledge of mathematics and other sciences to believe that “mathematics is not a science”, which I am sure you do not claim. Of course mathematics is a science, not an art, for example. Mathematics even turns out to be the best example of a rigorous science that humanity has. Thus whatever is interesting for mathematics is also interesting for science in general, since mathematics is a subset of science.

Sabine is very dogmatic... Just like the religious gurus she has become the very things she despises

Wrong.

Neh! Too many strong statements about "does not exist in nature" and "not relevant in science" to consider a good argument...

Thanks for clearing that one up!

LET ME SIMPLIFY TO FIRST PRINCIPLES: "Mathematics is a simple language (Grammar of continuous recursive disambiguation), consisting of one noun(Positional Name), one operation (Addition-Subtraction), and one agreement (Equality). This is the most reductive (simple) language possible. The combination of arbitrary correspondence (what the number refers to), and ordinality(position in the order), produce context independence and scale-independence. There are no context-independent and scale-independent measurements in physics (or in any aspect of reality). Likewise, this is why mathematics fails at the very small (quantum background), and very large (economics): because causal density is not reducible to a generalization in either case. Meaning all of mathematics is fundamentally statistical (reductive). As such, as we have complained, by criticizing Cantor for over a century, there are no infinities. Only unachievables and unknowns. In other words, never mix mathematical LANGUAGE (fiction) with physical MEASUREMENTS (description). The fact that mathematicians dont' know the foundations of mathematics as other than sets (language) and philosophers don't know that all sophistry consists of failure to respect the first rule of grammar (continuous recursive disambiguation to the point of unambiguity (identity)), is a symptom of one of the many catastrophic failures in the intelligentsia and academy in the pre-war period, and the 'credentialism' in the academy in the postwar period. There is no evidence that the number of people contributing to primary knowledge has increased since 1900. This should terrify us. And we should ask why. And this simple example, in this paragraph illustrates why: verbalism(idealism(sophistry)) vs action(existence (testimony)).

(And yes, I know this is word salad for those that don't understand the problem. There are however a number of us who do.)

I regret that I can only give this video one thumbs up. This opened my eyes to a better understanding of math and science. She made me smarter.

Your video proves the incompleteness of my understanding! Would you say that the uncertainty principle is Godel's incompleteness manifested in reality?

fire!

I read Godel's theorem. It immediately struck me that it only uses one definition per symbol in the "proof". This is at least wasteful, but immediately shows you the proof is incomplete since exhaustion of symbols is "the proof". Mark me down as unconvinced.

As per Wikipedia's note, one of the assumptions for the halting problem is wrong. There is no infinite computer. The function to return "halts or not" run on itself will eventually halt due to resource exhaustion (e.g. stack full). Otherwise you have a finite state machine, so for some small number of states the problem is solvable. For larger machines you can estimate the maximum time to a solution but not actually run it. So mark me down as unconvinced (per Wikipedia's note on this)

Please look up to "The Venus Project" it offers solution to most of the problems we face today.War, poverty, hunger, starvation, illnesses,unemployment, crime, deprivation, human suffering can be solved with THE VENUS PROJECT. It offers method to achieve better world.

Thank you. Mathematics is an imperfect tool to explain what we can not fully explain. Yes, I agree.

Mathematics is incomplete when defined within a formalist system. What is more accurate to say is that formalism is inconsistent and incomplete 🤘

In other words, when dealing with physics, use common sense. Very agreeable. Still I can imagine that there could be several counter arguments for this. Because things like time and space also bear infinity with them.

Stunning :)

Allegedly Godels incompatibility theory works on an infinite series thus it doesn't matter how many axioms you add

I think the chaotic nature of weather prediction is a perfect example of unpredictability. Even if you have a perfect weather model formula the predictability of it will always depend on the accuracy of the data and measurements you start with. Given that you can never make an infinitely accurate measurement there will always be limits to predicting things like weather.

Love the star trek esq dress. Gorgeous and smart. Love the videos!

"Mathemathics" 😄

Miss Sabine Hossenfelder ,just great. Even consider You as well as your voice..Singularity...

mathematics is a formalism that describe physical reality, is it best to think of it in more as description, representation or relation that can outline nature, physics and reality? It maybe a real but limited physical substructure and is more of the fundamental language of science.

wow that was a super clear summary of gödels theorem, going to memorize that one

Roger Penrose needs to watch this video and he might give up his "goedel proves consciousness is nonphysical" silliness lol

Sabine, what do you think about this result?
Undecidability of the Spectral Gap. Toby S. Cubitt, David Pérez-García and Michael M.
Wolf in Nature, Vol. 528, pages 207–211; December 10, 2015

@t

Does Gödel and Turing have some say about fundamental physics after all?

Has anyone predicted how many times I would go through this playlist?

If nothing real is infinite, why do we rely so heavily on Calculus which utilizes infinities?

Mathemathic lol

This makes presumptions on nature we do not know for certain, perhaps can not know for certain. But we should never attempt to justify resting on our laurels with potential problems like this. That IMHO is a dangerous scholarly precedent to set. Not saying you are, just a warning against laxity.

I like it how you keep pointing out that maths is not science. This fact seems to be getting more and more forgotten among many scientists, especially the theoretical ones.