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What is it like to understand advanced mathematics? (quora.com)
122 points by vinchuco 10 months ago | hide | past | web | favorite | 53 comments



There are many wonderful passages in the book "Genius" by James Gleick that describe Richard Feynman's deep intuitive understanding of calculus. He also talks about it a bit in his own works such as Surely You're Joking Mr. Feynman. Highly recommend if you want to see lots of interesting answers to this question. His key approach was that he always wanted to understand things based on "first principles". So he would usually take on a new mathematical frameworks by breaking them down and re-deriving them on his own from scratch.

Reminds me of how Ben Franklin taught himself to write by reading famous passages and then attempting to write them from memory, comparing his version to the original version afterward and seeing what he did differently and why.


Interestingly, this reads like the experience of expert knowledge in almost anything.

Like, I have expert knowledge in law, and I'm working toward at least more advanced (though far far from expert) knowledge in coding-things, and it very nicely matches how I interact with legal problems and can dimly see glimmers off at the end of a long tunnel in coding problems.

But at the same time, it's inspiring as all hell, because mathematics from the outside doesn't seem like the kind of thing that could ever produce the same satisfying and effective experiences of cognition.


It's like being really good at a video game that only you and maybe 2 other guys play.

You talk about the dungeons, monsters, level-bosses, etc but nobody can relate.


I somehow feel like this is written by Terence Tao, based on the quality, style, and length that are similar to what's on his blog. Anyone else shares the feeling or has a different guess?

Edit: In a later part, the answer says "Terence Tao is very eloquent about this here: ..." so likely not (based on factors like typical personality of a mathematician, but I could be wrong since I don't know him personally).



that later part could be a nice red herring for those who think it is Terry.


to understand advanced mathematics, is to start reading books on philosophy


Pretty much.

Mathematics is a language. Most people only learn arithmetic, and then only by rote. Sometimes, it is disheartening. I've had people who refused to accept that Ph.D. is Doctor of Philosophy.

I like to point out that nobody has successfully brought me a bucket of negative apples.

Err... I'm a mathematician. I am an applied mathematician. I am not smart enough to be a philosopher of mathematics, though I have my Ph.D.

Anyhow, math is a language that is capable of telling stories, truths, and even falsehoods.


As another fellow mathematician (albeit pure), I take some issue with your argument. Rather, I'd say the collection of apples is a clear manifestation of a monoid that is not a group. Why should all physical systems form a group? It seems almost intellectually dishonest to suggest that a particular non-example is evidence against the universality of a mathematical object.

Are symmetry groups not "physical" enough? I can rotate a piece of paper in the plane and then undo this operation just fine.

Algebraic and topological structures seem to manifest themselves readily within nature. I should think infinities provide greater concern.


>> Algebraic and topological structures seem to manifest themselves readily within nature. I should think infinities provide greater concern.

So basically what you are saying is that, we can't be so certain that we humans invented math. Math is there, structures were there to be discovered ? Why is there such a thing as monster group ? We named those things, but if we didn't exists, it would still be there in the abstract world ? This is what is bugging me lately, I can't decide if math was invented or not. Some parts may be invented, but some parts may be discovered.


I recommend Ian Stewart's Letters to a Young Mathematician. He addresses your question (Platonism vs formalism) pretty succinctly, in my opinion as a noob.

Accordingly "the working philosophy of most mathematicians is a mostly unexamined Platonist-Formalist hybrid ... Math is a product of human minds but not bendable to human will. Exploring it is like exploring a new tract of country ... but the mathematical countryside does not come into existence until you explore it."


I like to think that all theses things are already there as implied (by the chosen axioms) but unrealized potential, and then are made explicit by the act of exploring that potential (from a chosen point of view/using a chosen formalism).


The post may not have been clear enough, but the point is that it's impossible to really have negative apples. You can owe apples, but you can't physically possess them. Namely, pointing out how it remains a philosophical pursuit.

As for infinities... I'm not sure if they drove Cantor insane or if he was already insane. Some infinities are bigger than others, and I have strong opinions about infinity.

0.999... = 1

Horrible! ;-)


I disagree, I can physically possess 3 apples no more nor less than I can physically possess -3 apples.

To say "I have three apples" is merely a way of modeling a configuration of the world. The apples may be in my pocket, or in my hand, or in a bag I'm holding, or even on my desk in front of me. They may be in the barn out back, but in all cases we can say I have 3 apples. Saying "i have 3 apples" is a simplification of a ton of configurations of the state of the world where I have ownership of apples. Similarly I have -3 apples is a summary of all the configurations of the world where I owe three apples.

Neither is any more real than the other. And yes it is philosophical.


Not sure what that remark was in reference to, but ... yes, they can bring you negative apples.

Your enterprise can have an obligation to deliver apples to someone, which has all the same effects as negative apples and should be booked as such. If you acquired an enterprise with such an obligation, they gave you negative apples.

It's important to account for them that way in e.g. seeing the implications of the Put-Call Parity Theroem.


See, there's no such thing as negative apples. Sure, they exist in math - but they don't exist in reality. You can possess a chit that says you owe apples but, in reality, you only have said chit and do not actually possess negative apples.

The post may not have been clear enough, but the point was to show that it's still very much a philosophy.


> nobody has successfully brought me a bucket of negative apples.

But when they did I am going to take the square root of the number of them and break the universe.


I should have thought of this sooner. Since they're ALREADY talking about purely imaginary apples, take n of these. Arrange them in the bucket so as to make a square of side length n (of said imaginary apples). The result is n^2 negative apples and a really bad math joke.


No one would bring you a bucket of negative apples. Positive and negative ought to be thought of as directions, while the absolute value is the magnitude. A bucket of apples given to you has a positive direction (increasing your supply). A bucket of apples being taken from you has a negative direction (decreasing your supply).


Of course you can't bring me a bucket of negative apples. That's the point. It very much is a philosophy and doesn't always relate to reality. Reality is, you can never really have a bucket of negative apples.


I agree with you in part, many things in math have little direct relation to reality, or at least daily life for most people.

I just disagree with your example and I dislike it because I've literally had an argument with an engineer (an engineer!!!) about whether negative numbers were real who insisted on sticking to this sort of example. Negative indicates direction, this isn't really abstract. If it's increasing your balance of apples, it's positive. If it's decreasing your balance of apples, it's negative. Many people understand this quite well in the sense of monetary debt. I "possess" negative $xxxx, because that's the amount I owe someone else, in turn the holder of the debt "possesses" positive $xxxx because they expect to receive it at some point in the future.

If this is philosophy it's the barest levels, and it's philosophy that any culture with a concept of ownership and debt would have no difficulty with.


I was trained as an engineer and the abstract part of maths used to bug me a lot. We were often told to just accept them as they are, that just like everything else they add to your Toolbox. My biggest bugbear is imaginary numbers, can't seem to get my head around what they are but I accept that this is due to my limited imagination. Well I read that they can't be imagined because it's a human construct - honestly I don't know what to say or think.

Bucket of negative apples, hmm. I feel slightly sheepish to say this but I immediately visualised it to be "a bucket that you intend to put n apples in, but didn't." Thank god I'm in a semi-technical field now :D

Edit: language.


Which field of engineering?

In electrical engineering, imaginary/complex numbers are critical in circuit analysis once you move into the AC domain. You can try doing it with sines and cosines, but that's infeasible as soon as you move beyond anything basic. [1] But when you transform them into complex numbers using Euler's formula (one of the most beautiful things in maths, IMHO, especially Euler's identity) [2], it becomes "easy".

Then there's also the Fourier Transform [3] which transforms time-domain signals to frequency-domain.

So my visualisation of imaginary numbers is something involving circles and spirals being twisted and shifted all over the place. Hard to explain, but I've reached the point where I feel I have a good grasp on them. :)

[1] http://hyperphysics.phy-astr.gsu.edu/hbase/electric/impcom.h... [2] https://en.wikipedia.org/wiki/Euler%27s_formula [3] https://betterexplained.com/articles/an-interactive-guide-to...


Computer, mostly hardware (so heavy on electronics rather than programming)

Oh yes no doubt j is very useful - but that's what I meant by it being part of the Toolbox. To visualise it as an entity is something else though, and I know that you're not really supposed to do that, especially when it was 'invented' to be useful in the first place.

Perhaps with more experience and perseverance, I might have reached an understanding like yours. I do find myself regretting for giving up on engineering so soon, but hey, I can still take my time to understand what I couldn't before. Thanks for the links :)


You can and should visualize imaginary and complex numbers. Whoever told you otherwise was wrong. See my other post. Plot a complex number (say 2+4i) and then try two operations on it: multiply by 2+2i (this will rotate the point to the left and scale it); and try adding a complex number (say 1-3i) which will have the effect of translating the original point.

Developing an intuition about what rotation a particular complex number will give you is more complex, admittedly. But this is enough to visualize the effects, if not precisely mentally predict the result.


I was thinking about your post yesterday, thanks for it :) Sorry, I didn't clarify with what I meant by "visualising." When you think of an abstract entity, like a foreign word for example, you can sort of grasp what it means when you have it translated. But you can't do that with the imaginary number. You can only get a 'feel' of what it is when it's set in context, like the expressions you gave, and even then the application is still in a math world (geometric (?) space.)

In hindsight, I think that I approached it wrong, and it was far too simplistic anyway. I've been reading popular maths books lately and it's made me realise that there are more ways than just visualisation to understand something. Ironically (a happy one!) this is making me excited about maths for the first time in my life.


Imaginary numbers just have an unfortunate name. The complex numbers (having a real and imaginary component) are objects on a 2-dimensional plane. Using complex numbers lets you describe, in algebraic terms, geometric entities and operations. Adding a+bi is translation. Multiplying by i is rotation by 90˚ counterclockwise. Multiplying by a, a real valued scalar, is scaling. Multiplying by bi is rotation and scaling. Multiplying by a+bi is rotation and scaling again, but the angle of rotation will be something other than 90˚.

The complex number notation describes an object in space or an action to be performed on objects in space. Other than it being more compact, it's not substantively different than a statement describing an action like "travel 50m northeast, then turn left, then travel 40m".


Imagine a flying unicorn shooting rainbows out of it's face. I can imagine that, even though it's a human construction, so I reject the argument that you can't imagine human constructions.


But it's your fault no one can bring you this bucket: You want 'negative apple' entities but ask for (what is interpreted as) a negative number of apple entities.

A key distinction here is that one has a defined implementation. Making that distinction is philosophy. Talking within the framework of implementation is mathematics.


Right. It's very much a philosophy.

And it's not really my fault that nobody can bring me negative apples. It's not like I made the rules! Sheesh! ;-)


You did! You posed the question!

I just don't see the value in conflating both terms (philos and math) when there's a distinction pointed out above.


No, I didn't make the rules. Reality is what reality is. They don't let me decide the rules for reality. That's probably for the best.

You can't bring me a bucket of negative apples. You can bring representations of future apples, or something like that. However, negative apples do not actually exist. In all these years, nobody has brought me a basket of negative apples.


I don't contest that. But you have a choice to frame your question in a way conducive to a valid answer. But someone who doesn't do that is likely unworthy of negative apples anyways. :)


What if you're an apple vendor and someone gives you $5 bucks and an empty bucket, with the expectation that you're going to fill that bucket full of apples and give it back to them?

In the moment where you have an empty bucket and a debt of apples, I'd call that a bucket of negative apples.


No, you'd have a bucket devoid of apples, until you put apples into it. Then, you'd have fewer apples than you had before.

You can call it negative apples, if you want. However, it's just an empty bucket. ;-)


It's hilarious watching people struggle with this. The whole point is that you have to think about it - it depends on philosophy.

In other words, it's all in your head. A real, physical, "negative apple" does not exist in this world. Only the human philosophical concept of "owing someone apples" or similar such things. That's the guys' point. It doesn't matter whether you can bring representations of negative apples because those all depend on a philosophical bedrock.


“It is not the wind that moves, it is not the flag that moves, it is your minds that move.”


That PhD means "Doctor of Philosophy" is just an artifact of the fact that all doctoral degrees were once subcategories of "Philosophy".

Philosophy today is pretty useless.


"Philosophy today is pretty useless."

Studying philosophy teaches you how to think. If you consider that useless...


You claim that. Yet none of the most mentally demanding tasks were done by philosophers. Today philosophy is mostly meaningless intellectual posturing. In the past, it was the act of thinking without applying the scientific method. As Feynman said of Spinoza "meaningless chewing around".


If philosophy (and only philosophy) teaches people how to think, then who taught the first philosopher?


No one claimed only philosophy taught people how to think.


How do you know it doesn't just draw out a capacity to think that you already have, and that you could equally well have drawn out by studying or working at something else?


Maybe you could, but that still doesn't make philosophy useless.


I'd argue that philosophy may become more important than ever as we contemplate the ethics of handing decisions like who's going to die in a trolley-problem type self-driving car accident over to machines.


It'll be easier to (metaphorically) untie people from the tracks.

Program slow speed limits in dangerous areas, install sensor augmentation in areas where vehicle located sensors don't work well, rebuild roads to work better, etc.


Most philosophers throughout history would laugh at their body of knowledge being measured in terms of utility. And then would become saddened that economics has become the dominant global value system.


Economics is the word for the dominant global value system. I think a sad philosopher has more work to do.


A bit parochial to say that Philosophy--as it is taught or studied today--is useless, in my opinion. You're entitled to that thought but you might be surprised by the utility Philosophy has to offer.

I think Philosophy studied with other disciplines is a particularly potent combination. Of my friends who have done this (they are very few) they are perhaps the most inspiring and trenchant thinkers I know.


I think it's sad that your view that philosophy is worthless has become more of a mainstream feeling than it ought to be. Not everything needs to be measured in terms of usefulness. Interestingly enough, I think you find many developers have a natural affinity for philosophy. All this talk about AI, for example, is just philosophy.


It must be extremely difficult for you to drive around in your car that is unable to go in reverse.

Math is fascinating. It is also just a tool.


For someone who has forgotten math (other than high school level) and never did philosophy, what books/courses are good to start with (especially philosophy)?


The community college course I took past spring used Thomas Nagel (What Does It All Mean?) supplemented with some handouts. I would recommend a class because part of the doing of philosophy is the discussion. Plus the classroom is one of the few places where you can delve into the nuances of an issue without being considered pedantic or boorish.




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