- https://youtu.be/EbzESiemPHs?si=4UNA7JGPt7OmfnOi&t=206
Here's Gromov, one of the greatest geometers of the last 50 years, discussing his viewpoint on this.
- Simons himself completely disspells this idea in his interview on Numberphile.
- The EMH is a description of how the market behaves when a sufficiently large number of independent actors are looking for alpha. It is not a prescription of how the market should behave.
The conclusion is that with a sufficiently large number of actors in the market all seeking profits by trying to find misevaluation of stock prices, the excess profits of any individual actor will (assuming they all have access to the same information) converge to zero.
Its less a paradox and more a matter of game theory. Every investment firm which gives up trying to look for alpha (believing it is fruitless) means the remaining firms have more opportunities to find stocks with available information not reflected in the price. There's no paradox here: each individual actor is incentivized to participate in order to not miss out on that potential for excess profits, and the net effect is the EMH.
- The practical experience of doing mathematics is actually quite close to a natural science, even if the subject is technically a "formal science* according to the conventional meanings of the terms.
Mathematicians actually do the same thing as scientists: hypothesis building by extensive investigation of examples. Looking for examples which catch the boundary of established knowledge and try to break existing assumptions, etc. The difference comes after that in the nature of the concluding argument. A scientist performs experiments to validate or refute the hypothesis, establishing scientific proof (a kind of conditional or statistical truth required only to hold up to certain conditions, those upon which the claim was tested). A mathematician finds and writes a proof or creates a counter example.
The failure of logical positivism and the rise of Popperian philosophy is obviously correct that we can't approach that end process in the natural sciences the way we do for maths, but the practical distinction between the subjects is not so clear.
This is all without mention the much tighter coupling between the two modes of investigation at the boundary between maths and science in subjects like theoretical physics. There the line blurs almost completely and a major tool used by genuine physicists is literally purusiing mathematical consistency in their theories. This has been used to tremendous success (GR, Yang-Mills, the weak force) and with some difficulties (string theory).
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Einstein understood all this:
> If, then, it is true that the axiomatic basis of theoretical physics cannot be extracted from experience but must be freely invented, can we ever hope to find the right way? Nay, more, has this right way any existence outside our illusions? Can we hope to be guided safely by experience at all when there exist theories (such as classical mechanics) which to a large extent do justice to experience, without getting to the root of the matter? I answer without hesitation that there is, in my opinion, a right way, and that we are capable of finding it. Our experience hitherto justifies us in believing that nature is the realisation of the simplest conceivable mathematical ideas. I am convinced that we can discover by means of purely mathematical constructions the concepts and the laws connecting them with each other, which furnish the key to the understanding of natural phenomena. Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be deduced from it. Experience remains, of course, the sole criterion of the physical utility of a mathematical construction. But the creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed. - Albert Einstein
- >Today, mathematics is regarded as an abstract science.
Pure mathematics is regarded as an abstract science, which it is by definition. Arnol'd argued vehemently and much more convincingly for the viewpoint that all mathematics is (and must be) linked to the natural sciences.
>On forums such as Stack Exchange, trained mathematicians may sneer at newcomers who ask for intuitive explanations of mathematical constructs.
Mathematicians use intuition routinely at all levels of investigation. This is captured for example by Tao's famous stages of rigour (https://terrytao.wordpress.com/career-advice/theres-more-to-...). Mathematicians require that their intuition is useful for mathematics: if intuition disagrees with rigour, the intuition must be discarded or modified so that it becomes a sharper, more useful razor. If intuition leads one to believe and pursue false mathematical statements, then it isn't (mathematical) intuition after all. Most beginners in mathematics do not have the knowledge to discern the difference (because mathematics is very subtle) and many experts lack the patience required to help navigate beginners through building (and appreciating the importance of) that intuition.
The next paragraph about how mathematics was closely coupled to reality for most of history and only recently with our understanding of infinite sets became too abstract is not really at all accurate of the history of mathematics. Euclid's Elements is 2300 years old and is presented in a completely abstract way.
The mainstream view in mathematics is that infinite sets, especially ones as pedestrian as the naturals or the reals, are not particularly weird after all. Once one develops the aforementioned mathematical intuition (that is, once one discards the naive, human-centric notion that our intuition about finite things should be the "correct" lens through which to understand infinite things, and instead allows our rigorous understanding of infinite sets to inform our intuition for what to expect) the confusion fades away like a mirage. That process occurs for all abstract parts of mathematics as one comes to appreciate them (expect, possibly, for things like spectral sequences).
- This is related to Terence Tao's notion of the stages of mathematical rigor.
As Tao puts it, the value of intuition becomes much higher in the post-rigorous stage once you have sufficiently developed your technical skills.
https://terrytao.wordpress.com/career-advice/theres-more-to-...
- I certainly am not making any comments about her experiences for sure! Academia is difficult and full of terrible stories, and its not surprising that it causes many people to become exceedingly bitter and contrarian (Peter Woit is famously of the same ilk as another string theorist critic who fell out of academia like Sabine).
Unfortunately a chip (even a legitimately earned one) on ones shoulder about the bad parts of academia doesn't save you from being criticized for being crank-y.
- As an expert in at least some of the things Sabine makes videos about (string theory), Sabine is a contrarian who, if you are not otherwise an expert on what she is talking about, it would be best to avoid.
Sabine, like many contrarians, takes advantage of the fact that there are smart and convincing criticisms of many mainstream ideas, and she does her best to rely on those criticisms. However like all contrarians she presents a biased and exaggerated view of things in order to stoke engagement, and unless you are an expert it can be difficult/impossible to determine whether the view she is giving is balanced.
This is a classic issue with string theory critics, because string theory has many legitimate problems with it, but many of the critics are intellectually dishonest and you probably shouldn't listen to their criticisms on principle (but even I must admit it's quite hard to find good quality intellectually honest criticism of string theory which is digestible, so these contrarians tend to be the only loud voice).
In Sabine's case it is not so bad, because it is clear from some of her other positions that she is basically a crank. MOND and superdeterminism are basically crank physics at this point but she supports them purely because she is a contrarian. On this evidence alone you should not trust anything she says on any other subject, otherwise you're falling for a kind of Gell-Mann amnesia.
- Wildberger is a crank
- A theorem which is true in every model is provable by Godel's completeness theorem. Since this theorem is true for the standard model of the natural numbers but not provable, it follows there are nonstandard models of the natural numbers for which it is false.
That is, there are models of Peano arithmetic which contain all of the natural numbers we know and love, and some other ones on top of that and there are some Goodstein sequences using those extra "non-standard" natural numbers which do not terminate at zero.
https://en.wikipedia.org/wiki/Non-standard_model_of_arithmet...
- The actual answer is the assumptions which define a self-propagating wave do not apply once the wave leaves a vacuum. When it becomes incident onto some medium, due to the coupling of electrons within the medium to the electromagnetic field, the pure electromagnetic wave gets transformed into a phonon, which is a combination of electromagnetic and mechanical oscillation within the medium (and therefore has speed <c, depending on the particular properties of the medium). When the phonon subsequently leaves the system, those traveling oscillations induce a new self-propagating wave on the other side, sending the light on its way as usual.
- One of two things:
Either what we know about black hole formation is basically complete (it goes gas -> star -> black hole -> accretion + collisions) but the environment in the early universe was sufficiently different/dense that parameters which rule out the formation of supermassive black holes now were different. Maybe there were many intermediate black holes just in the millions of years after the big bang and things were still close enough together that accretion could happen and collisions were "likely" at the rate needed to form SMBHs after just a billion years. If that is true we might expect to see many many active galactic nuclei as we get better telescopes and look further back, depending on how quickly such black holes formed.
The other option is there is a mechanism of black hole formation that bypasses the above chain which we understand. People talk about supermassive stars, gas clouds collapsing directly into black holes, or primordial black holes that existed due to essentially random distributions of density moments after the big bang causing some regions of space to collapse into massive black holes which then persisted. Such things are far more difficult to observe, but could be inferred if we don't see many many active nuclei as we get better telescopes but all other indications of the accuracy of the big bang + inflationary theory hold true.
- Also important to note that the process of stellar collapse and then black hole accretion takes absolutely enormous amounts of time to collate a large amount of mass together. It's also an extremely energetic process, you would expect to see very bright black holes if millions of solar masses of matter were infalling creating very large and bright accretion disks. We do see some active galactic nuclei but not that many. There's just no way there was enough time for this to happen in the early universe, or really even after a measly 14 billion years (i.e. seeing these young supermassive black holes is challenging for the stellar collapse theory, but the theory was already pretty challenged).
Not to mention if supermassive black holes were being formed by accretion, you would expect to see many intermediate mass black holes (1000-1000000 solar masses) everywhere, but we see almost none.
- On that of course I agree, but mathematicians tend to "relegate" such things to exercises. This tends to look pretty bad to enthusiasts reading books because the key examples aren't explored in detail in the main text but actually those exercises become the foundation of learning for people taking a structured course, so its a bit of a disconnect when reading a book pdf. When you study such subjects in structured courses, 80%+ of your engagement with the subject will be in the form of exercises exploring exactly the sorts of things you mentioned.
- > I’m sure it’s important if you want to be a mathematician, but if you just want to understand enough to be able to use it?
This book is for people who want to be mathematicians.
- Mathematicians are well aware of complaints like these about introductions to their subjects, by the way.
It is for a reason that this book introduces the theory of abstract vector spaces and linear transformations, rather than relying on the crutch of intuition from Euclidean space. If you want to become a serious mathematician (and this is a book for such people, not for people looking for a gentle introduction to linear algebra for the purposes of applications) at some point it is necessary to rip the bandaid of unabstracted thinking off and engage seriously with abstraction as a tool.
It is an important and powerful skill to be presented with an abstract definition, only loosely related to concrete structures you have seen before, and work with it. In mathematics this begins with linear algebra, and then with abstract algebra, real analysis and topology, and eventually more advanced subjects like differential geometry.
It's difficult to explain to someone whose exposure to serious mathematics is mostly on the periphery that being exposed forcefully to this kind of thinking is a critical step to be able to make great leaps forward in the future. Brilliant developments of mathematics like, for example, the realisation that "space" is an intrinsic concept and geometry may be done without reference to an ambient Euclidean space begin with learning this kind of abstract thinking. It is easy to take for granted the fruits of this abstraction now, after the hard work has already been put in by others to develop it, and think that the best way to learn it is to return back to the concrete and avoid the abstract.
- Realistically an AI model is basically just a very complicated piece of software. The model weights are akin to the software code, the model outputs are akin to the outputs a user of the software creates, and the datasets are akin to the intellectual property put into the software by the developer to create the code.
In the same way that a developer could not simply steal someone elses intellectual property in order to develop a feature of a piece of software, one cannot simply steal the intellectual property to adjust the model weights. The main difference is its generally quite easy to see in practice if a model has utilized some intellectual property (because for example you can ask ChatGPT to recite the first 100 words of Harry Potter) compared to another piece of software where you'd need access to the source code or developers thoughts (which could only be achieved through litigation, in most circumstances).
I think a great many people come up with convoluted answers to this question because they are uncomfortable with the reality that these very large organizations have essentially stolen hoards of intellectual property, and now that the horse has bolted people want to justify not closing the barn door. It seems to me very simple: to train an AI model on data, you must respect its copyright. The model weights should be copywriteable by the developers of the model (even if the law currently does not allow this), and the outputs of the model should be copywriteable by the person who interacted with the model (software) to produce the outputs.
The analogy with Photoshop is extremely simple: If some other software invented Gaussian blurring and copywrited it, then Adobe would have to license that technology from them to include it as a feature in Photoshop. The actual photoshop software/code would be copywrited by Adobe, and if someone created an blurry image with Photoshop they can copywrite it.
I think people only disagree with this due to some sense that the process of translating data to model weights is "automatic" or "computational" in nature. You could in principle get a person to, by hand, go through millions of data sets and compute the changes to the model weights. This is no different to someone writing a piece of code, checking someone elses approach, and adjusting their own code after the fact. It just happens that we have developed very effective tooling to automate the adjusting of the code.
most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to formalism https://en.wikipedia.org/wiki/Mathematical_Platonism
Mathematicians begrudgingly retreat to formalism and foundations when pressed because its easier to defend, but the day-to-day of contemporary mathematics is much more an explorative process of a "real" mathematical landscape. They aren't concerned with foundations because it "feels" self-evident that the mathematics they are discovering is true (because their means of discovery, rigour and proof, "guarantee" it to be so).
A lot of the comments here are making false assumptions like "but surely mathematicians all know that their field is ultimately justified as a symbol-pushing game from some axiomatic system right?" in the same way one might say "surely all computer scientists know that every language ultimately compiles down to 1s and 0s processed by a CPU" but that is not at all how most mathematicians think about doing mathematics.