Willingness to look stupid and intellectual self-confidence are two sides of the same coin.
If you can find internal (rather than external) reasons to trust/believe in your own intelligence and capabilities, it makes it easier to be willing to look foolish. Also, a lack of knowledge/ability in a new area (or even a familiar area) is not a sign of a lack of capability. There's a difference between being a novice and being an idiot. So long as your source of intellectual self-confidence is strong enough (say, you have made great intellectual achievements in some other area of your life unrelated to the thing you're struggling with right now) its irrelevant if other people think you the fool: they're simply mistaken, and that's no skin off your back.
Someone (supposedly) published da Vinci's to do list a while back, and from the snippets I read he seemed to spend most of his time talking to experts about subjects he didn't know much about. Pretty telling if true.
Something I was told, very luckily I now believe, early in life is that for the most part we are all as smartish as each other, the only real difference is how much attention & time you're willing to pay to a particular thing(s). Yes, it's glib but I continue to believe it holds more truth than many of us would like to believe
You need to differentiate between special and general relativity when making these statements.
It is absolutely true that someone else would have come up with special relativity very soon after Einstein. All that would be necessary is someone else to have the wherewithal to say "perhaps the aether does not need to exist" for the equations already known at the time by others before Einstein to lead to the general theory.
General relativity is different. Witten contends that it is entirely possible that without Einstein, we may have had to wait for the early string theorists of the 1960s to discover GR as a classical limit of the first string theories in their quest to understand the strong nuclear force.
As opposed to SR, GR is one of the most singular innovative intellectual achievements in human history. It's definitely "out of distribution" in some sense.
> 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.
This is his argument that mental experiments are a good basis to construct a theory. If relativity had 0 experimental evidence to this day it would be held in considerably lower regard.
Peter Woit, the Columbia maths department computer systems administrator, makes his bread by googling the word String Theory and then posting what ever latest results come up in a disingenuous way on his blog to stir reactions from his readers.
Because, unfortunately, a terminal degree is not always a guarantee of future productivity.
String theory came into the public consciousness about two decades ago, followed by a backlash from a relatively small but persistent set of professors. String theory has largely dropped out of the public focus, but those researchers continue to hammer at it.
Woit is one of those. He's known primarily for his objections to string theory. His own work is just fine, but nobody outside of his discipline knows it.
Something the computer scientists of Hackernews might not realise is that most mathematicians are by nature Platonists, even if they would not try to defend that position when pressed.
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.
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.
Yeah, I think the "paradox" is usually a problem for pundits and academics and not practitioners. Lots of people have experience finding and correcting market inefficiencies, usually getting paid for it.
Yeah, I feel like people have this idea that the EMH is 'economists think markets are perfectly efficient' when really it's 'under these idealised conditions a market should approach perfect efficiency' and any real market is obviously not going to be perfectly efficient, but ones that get closer to those conditions should be more efficient.
(And looking at how traders work, it's all about finding a strategy that no-one else has found and executing on it. Once two competitors with similar resources know the strategy, it ceases to be particularly profitable, which to me seems to be pretty in line with the EMH)
If the market is efficient then there is no risk adjusted alpha, which renders the search for it a waste of time and effort, which means no actor would rationally continue it, which means there is no mechanism for price discovery, which would render the market totally inefficient.
This is the paradox.
EMH is unfalsifiable at best and tautological at worst.
>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).
> Pure mathematics is regarded as an abstract science, which it is by definition.
I'd argue that, by definition, mathemtatics is not, and cannot be, a science. Mathematics deals with provable truths, science cannot prove truth and must deal falsifiability instead.
You could turn the argument around and say that math must be a science because it builds on falsifiable hypotheses and makes testable predictions.
In the end arguing about whether mathematics is a science or not makes no more sense than bickering about tomates being fruit; can be answered both yes and no using reasonable definitions.
> In the end arguing about whether mathematics is a science or not makes no more sense than bickering about tomates being fruit
That's the thing, though — It does make sense, and it's an important distinction. There is a reason why "mathematical certainty" is an idiom — we collectively understand that maths is in the business of irrefutable truths. I find that a large part of science skepticism comes from the fundamental misunderstanding that science is, like maths, in the business of irrefutable truths, when it is actually in the business of temporarily holding things as true until they're proven false. Because of this misunderstanding, skeptics assume that science being proven wrong is a deathblow to science itself instead of being an integral part of the process.
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).
————
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
An alternative to abstraction is to use iconic forms and boundary math (containerization and void-based reasoning). See Laws of Form and William Bricken's books recently. Using a unary operator instead of binary (Boolean) does indeed seem simpler, in keeping with Nature. Introduction: https://www.frontiersin.org/journals/psychology/articles/10....
Mathematical "truth" all depends on what axioms you start with. So, in a sense, it doesn't prove "truth" either - just systemic consistency[1] given those starting axioms. Science at least grapples with observable phenomena in the universe.
Mathematical proofs are checked by noisy finite computational machines (humans). Even computer proofs' inputs-outputs are interpreted by humans. Your uncertainty in a theorem is lower bounded by the inherent error rate of human brains.
I agree we can't be absolutely certain of anything (maybe none of our memories are real and we just popped into existence etc.)
But we can be more sure of the deductive validity of a proof than we can be of any of the claims you make in these sentences, so I don't think they can serve to establish any doubt. If we're wrong about deductive logic, then we can only be more wrong about any empirical claims, which rely on deductive logic plus empirical observations
Plenty of mathematical proofs have been proven true with 100% certainty. Complicated proofs that involve a lot of steps and checking can have errors. They can also be proven true if exhaustively checked.
This may be, but not, I think, in a way that is particularly worth modeling?
When we try to model something probabilistically, it is usually not a great idea to model the probability that we made an error in our probability calculations as part of our calculations of the probability.
Ultimately, we must act. It does no good to suppose that “perhaps all of our beliefs are incoherent and we are utterly incapable of reason”.
You're saying maybe people have mistakenly accepted incorrect proofs now and again, so some theorems that people think are proven are unproven. I agree that this seems very likely.
In practice when proofs of research mathematics are checked, they go out to like 4 grad students. This isn't a very glamorous job for those grad students. If they agree then it's considered correct...
But note this is just the bleeding edge stuff. The basic stuff is checked and reproven by every math undergrad that learns math. Literally millions of people have checked all the proofs. As long as something is taught in university somewhere, all the people who are learning it (well, all the ones who do it well) are proving / checking the theory.
Anyway, when the scientific community accepts a bad proof what effectively happens is that we've just added an extra axiom.
Like when you deliberately add new axioms, there are 3 cases
- Axiom is redundant: it can be proven from the other axioms. (this is ... relatively fine? we tricked ourselves into believing something that is true is true, the reason is just bad.)
This can get discovered when people try to adapt the bad proof to prove other things and fail.
Also people find and publish and "more interesting", "different" proofs for old theorems all the time. Now you have redundancy.
- Axiom contradicts other axioms: We can now prove p and not p.
I wonder if this has ever happened? I.e. people proving contradictions, leading them to discover that a generally accepted theorem's proof is incorrect. It must have happened a few times in history, no?
o/c maybe the reason this hasn't happened is that the whole logical foundation of mathematics is new, dating back to the hilbert program (1920s).
There are well known instances of "proofs" being overturned before that, but they're not strictly logically proofs in the hilbert-program sense, just arguments. (Of course they contain most of the work and ideas that would go into a correct proof, and if you understand them you can do a modern proof)
Cauchys proof that, if a sequence of continuous functions converges [pointwise] to a function, the limit function is also continuous (cauchys proof only holds for uniform convergence, not pointwise convergence - but people didnt really know the difference at the time)
- Axiom is independent of other axioms: You can't prove or disprove the theorem.
English doesn't have a "I'm just hypothesizing all of this" voice, if it did exist this post should be in it. I didn't do enough research to answer your question. Some of the above may be wrong, e.g. the part about the 4 grad students.
One should probably look for historical examples.
Mathematics is a science of formal systems. Proofs are its experiments, axioms its assumptions. Both math and science test consistency—one internally, the other against nature. Different methods, same spirit of systematic inquiry.
It's not an empirical science, but it is a science, where "science" means any systematic body of knowledge of an aspect of a thing and its causes under a certain method. (In that sense, most of what are considered scientific fields are families of sciences.) Mathematics is what you'd call a formal science with formal structure and quantity as its object of study and deductive inference and analysis as its primary methods (the cause of greatest interest is the formal cause).
A proof is just an argument that something is true. Ideally, you've made an extremely strong argument, but it's still a human making a claim something is true. Plenty of published proofs have been shown to be false.
Math is scientific in the sense that you've proposed a hypothesis, and others can test it.
Difference is mathematical arguments can be shown to be provably true when exhaustively checked (which is straight forward with simpler proofs). Something you don't get with the empirical sciences.
Also the empirical part means natural phenomena needs to be involved. Math can be purely abstract.
You're making a strong argument if you believe you checked every possibility, but it's still just an argument.
If you want to escape human fallibility, I'm afraid you're going to need divine intervention. Works checked as carefully as possible still seem to frequently feature corrections.
The difference is that in mathematics you only have to check the argument. In the empirical sciences you have to both check the argument and also test the conclusion against observations
Empirical science uses both deductive logic to make predictions, and observations to check those predictions. I'm not saying that's all it involves. Not sure which part of that you disagree with
And a lot of what goes on in foundations of mathematics could be described as "testing the axioms", i.e. identifying which theorems require which axioms, what are the consequences of removing, adding, or modifying axioms, etc.
Somewhat tangential to the discussion: I have once read that Richard Feynman was opposed to the idea (originally due to Karl Popper) that falsifiability is central to physics, but I haven't read any explanation.
I'm not sure if it deals only with provable truths? It even deals with the concept of unprovability itself, if the incompleteness theorem is considered part of mathematics
Yes, but Godel proved the incompleteness theorem, by ingeniously finding ways to prove things about unprovability.
The incompleteness theorem doesn't say that there are statements which are unprovable in any absolute sense. What it says is that given a formal system, there will always be statements which that particular formal system can't prove. But in fact as part of the proof, Godel proves this statement, just not by deriving it in the formal system in question (obviously, since that's what he's proving is impossible).
The way this is done is by using a "metalanguage" to talk about the formal theory in question. In this case it's a kind of ambient set theory. Of course, the proof also implies that if this ambient metalanguage is formalized then there will be sentences which it can't prove either, but these in general will be different sentences for each formalized theory.
Science involves both deductive and inductive reasoning. I would in turn argue that mathematics is a science that focuses heavily (but not entirely) on deductive reasoning.
He probably means science in a wider sense as opposed to the anglo-american narrower sense where science is just physics, chemistry, biology and similar topics.
Who cares? That's just semantics. If we define science as the systematic search for truths, then mathematics and logic are the paradigmic sciences. If we define it as only empirical search for truth then perhaps that excludes mathematics, but it's an entirely unintersting point, since it says nothing.
> Euclid's Elements is 2300 years old and is presented in a completely abstract way.
depends on what you mean by completely abstract. Euclid relies in a logically essential way on the diagrams. Even the first theorem doesn't follow from the postulates as explicitly stated, but relies on the diagram for us to conclude that two circles sharing a radius intersect.
Not only is intuition important (or the entire point; anyone with some basic training or even a computer can follow rules to do formal symbol manipulation. It's the intuition for what symbol manipulation to do when that's interesting), but it is literally discussed in a helpful, nonjudgmental way on Math Stack Exchange. e.g.
Other great sources for quick intuition checks are Wikipedia and now LLMs, but mainly through putting in the work to discover the nuances that exist or learning related topics to develop that wider context for yourself.
> 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.
I may be off-base as an outsider to mathematics, but Euclid’s Elements, per my understanding, is very much grounded in the physical reality of the shapes and relationships he describes, if you were to physically construct them.
Quite the opposite, Plato, several hundred years before Euclid was already talking about geometry as abstract, and indeed the world of ideas and mathematics as being _more real_ than the physical world, and Euclid is very much in that tradition.
I am going to quote from the _very beginning_ of the elements:
Definition 1.
A point is that which has no part.
Definition 2.
A line is breadthless length.
Both of these two definitions are impossible to construct physically right off the bat.
All of the physically realized constructions of shapes were considered to basically be shadows of an idealized form of them.
Another point to keep in mind is that a lot of mathematics that's not considered abstract _now_ was definitely considered "hopelessly" abstract at the time of its conception.
The complex number system started being explored by the greeks long before any notion of the value of complex spaces existed, and could be mapped to something in reality.
I don't think we can say the Greeks were exploring complex numbers. There's something about Diophantus finding a way to combine two right-angled triangles to produce a third triangle whose hypotenuse is the product of the hypotenuses of the first two triangles. He finds an identity that's equivalent to complex multiplication, but this is because complex multiplication has a straighforward geometric interpretation in the plane that corresponds to this way of combining triangles.
There's a nice (brief) discussion in section 20.2 of Stillwell's Mathematics and its History
Plato was only about a generation before Euclid. Their lives might have even overlapped, or nearly so: Plato died in 347BC and Euclid's dates aren't known but the Elements is generally dated ~300BC
The only things that are weird in math are things that would not be expected after understanding the definitions. A lot of the early hurdles in mathematics are just learning and gaining comfort with the fact that the object under scrutiny is nothing more than what it's defined to be.
If you can find internal (rather than external) reasons to trust/believe in your own intelligence and capabilities, it makes it easier to be willing to look foolish. Also, a lack of knowledge/ability in a new area (or even a familiar area) is not a sign of a lack of capability. There's a difference between being a novice and being an idiot. So long as your source of intellectual self-confidence is strong enough (say, you have made great intellectual achievements in some other area of your life unrelated to the thing you're struggling with right now) its irrelevant if other people think you the fool: they're simply mistaken, and that's no skin off your back.
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