As a counterpoint to the overly exuberant article let me chime in as a graduating theoretical computer scientist. Math is hard and despite its beauty and allure academic life doesn't take place in a vacuum- ergo there's a lot of politics and pettiness from your peers. You should expect to invest years of your life into making progress on some hard problem with little encouragement in the meantime, this takes a tremendous toll on the mind and not for someone who falls easy prey to self doubt. Sometimes even after you publish your top result that took a lot out of you, it may take many years before people appreciate, let alone understand, your work- the vast majority of papers are never going to be read. To compensate for this horrible feedback mechanism you need to basically play the popularity game and try to give many talks and talk up your result when you meet people, so there's a lot of salesmanship involved here as well. So the career path of a junior scientist is pretty crushing mentally and I couldn't stomach it in the long run.
In an ideal world I might have continued in academia but the career path is so twisted you have to either be insanely good (at math and at managing time) or just hate yourself enough to sacrifice your best years working essentially in the metaphorical darkness well outside the spotlight and most likely alone and poorly paid.
I can't find the citations right now, but there was a recent study done through Elsiver's website. The group got a hold of the servers for Elsiver and took a look at page views. They wanted to know what the rates of papers being read was. It was... disheartening. They found that ~46% (again, not sure here as I can't find the source) of papers will never be looked at outside of the authors, reviewers, and the editors. The articles are not only never downloaded, but the pages are never even loaded. The stats were, to me, confusing, but hey that is what peer review is for. Still, if you take this paper (that I can't find now) to be true, ~half of all papers are effectively lecturing into the void. I admit, I drank a bit after reading that one.
In CS, most authors host "preprint" copies of their papers on their own website, and most views are of those versions. So I wouldn't be surprised if Elsiver's defunct distribution system isn't getting used much (at least in this field).
Yes, but roughly half of all papers will never even be page-loaded from the 'official' source (if you believe my recollection). I am not a CS researcher, but I would imagine that of 10 people that cite a paper, at least one will bother to download the original paper or look up the source. Hell, even 'vanity' page-views of your own work would have been counted in the paper that I mentioned, and that is with many authors on a single paper too.
I honestly don't know what to make of it really. I at least bother to look at my papers once, if only to show my family on the Holidays, and I have a lot of so-authors that may be doing the same. Are most researchers so fed-up with their own work as to not even bother looking at it again? What is going through their minds concerning their efforts? It's just ... heartbreaking. At least half of researchers don't seem to care at all, not even the people that put all that time and effort to get their research out there. Like, what are we doing with our lives?
To be fair most papers are written for readers that have the exact same viewpoint and specialty. So they are incredibly difficult to parse for people in the same field but a different sub specialty.
I think many more papers would be read if authors invested more time in learning how to write.
Isn't that plausibly the result of publish or perish? I would imagine it would lead to people publish at the cadence of their typing rate (to be cruel), not at the rate which they can generate good ideas.
Higgs said he could have not been able to operate in todays academia. Not everyone needs to have John von Neumann performance levels to do valuable work.
IIRC a lot of that has to do with spammy 'publication mill' journals. If you just look at 'decent' journals, the situation doesn't look nearly as dire.
Yes, Elsevier is far from a journal mill designed to H-index hack your way up the credential ladder. They are scum, yes, but they are at least truthful.
Maybe if Elsevier was open source & easily accessible more people would read it. I have tried reading many journal papers as a layman but in most cases got blocked by a pay wall. I dont like to rely on newspaper headline but like to go the source and because of paywall I get thwarted.
In addition, you can just email the authors too. Most, if not all, are very happy to send you a copy and help explain things too. Most scientists do want to talk about their work!
Elsevier used to demand you sign over all copyright, which would make that illegal (think about it...!!) They've also sent takedown notices to academics that posted their own articles on their own website. Oh, and by the way, Elsevier also sued Sci-hub and LibGen...
My own experience: When I have been on a University campus' wifi and needed to look at a paper, I could download and see the entire article, not just the abstracts. This has been true for all of the limited number of times I have tried this. I have also done the same at some, but not all, conferences; typically the very large ones only. I would then doubt that the numbers are being skewed towards the scihub and other 'rebellious' forms of paper acquisition. However, this is only my experience.
Maybe I skimmed the article, but my takeaway from this article was that using our abilities to think and solve mathematical problems helps us to live a life of more intellectual resilience.
In my view, this wasn't about doing mathematics as an academic career, climbing up the ivory tower, publish or perish, and all that.
Edit:
I tend to agree.
My intellectual capacity was far improved only after doing first year Engineering Maths & Physics. I did terrible gradewise, but it has helped me visualize, juxtapose mental structures, quickly iterate on different concepts etc.
I should go back to doing something again. Had to do some cartesian products in a data migration the other day, in my dayjob, it was refreshing :).
And as always, this timeless quote from Einstein motivates
"Do not worry too much about your difficulties in mathematics, I can assure you that mine are still greater."
> My intellectual capacity was far improved only after doing first year Engineering Maths & Physics. I did terrible gradewise, but it has helped me visualize, juxtapose mental structures, quickly iterate on different concepts etc.
Same here, and thanks to the internet just remembering concepts is usually enough these days.
Yesterday I had to quickly guestimate for a project how much space overhead we'd have on the server if we pre-generated an image pyramid of tiles for leaflet.js[0], instead of generating a "PNG" in RAM on the fly each time and serving that. Then I realised the number of pixels shrinks by a quarter every time we zoom out, so the worst case would be the infinite sum of 1/4^n. While I vaguely remember from my first year of physics how to calculate that, it's been over a decade. But we have Wolfram Alpha these days[0], which told me it converges on 4/3. So I knew the upper boundary would be around 4/3 of our original data, plus negligible PNG header overhead, and assuming compression is about the same.
Bachelor of Science in Mechanical Engineering with minors in Math and Physics here...
While I can probably count on one hand the times I've used what I learned in the "real" world, I can say I really value my education.
I believe the real value of a math based curriculum or even something like philosophy with an emphasis in logic is that it teaches one how to think in a reasoned, dispassionate and rational way.
Doing serious research level mathematics (or any other science for that matter) is almost impossible outside academia. Doing serious work is hard and mentally draining. Mere mortals can't motivate themselves to do it unless the incentives align just right (i.e. being in academia). Also it takes all your time so forget about it if you have another job.
I know this from personal experience. After I joined industry I thought I'll do physics research in my own time. In fact, I can't even keep myself up to date about current research. Most journal articles require a lot of effort to go through, and I don't have that kind of energy after a work day.
In math we have lots of journals devoted to recreational and more elementary problems. There is a lot of low hanging fruit out there.
For instance, compute the fibbonacci numbers modulo the square of a prime and compare to modulo a prime. Are there numbers where the first zero appears at the same point? We don't know.
I'm not saying the problem is easy. I'm saying one can make some progress without much theory.
> I'm saying one can make some progress without much theory.
Claimed without evidence.
Sure, you can make trivial progress by writing a computer program to search, and the problem is "elementary", but "elementary" doesn't mean "can be solved without studying lots of techniques and theories", it means "can be solved without calculus"
I know the feeling, but don't forget that 'human flourishing' part of your life:
... the achievement of human flourishing, a concept the ancient Greeks called eudaimonia, or a life composed of all the highest goods. Su talked of five basic human desires that are met through the pursuit of mathematics: play, beauty, truth, justice and love.
Math is different. Even complete outsiders of mortal cognitive gifts can now and again generate publishable results.( I tried to find the reference but my google-fu fails me now - I do have actual published papers in mind.)
In physics these laymen are generally called 'crackpots' for a good reason because the context one needs to understand to generate new results is so huge.
Mathematics is different - all one needs is pen, paper, a mind that's attuned to generating logically whole statements and a problem that just won't let go of ones mind. The results are likely in geometry or some other approachable field and not in some more obscure subject.
That definition is circular. Of course research-level mathematics is done in academia, but there is a much larger field of applied mathematics that is done in industry.
That stuff has very little to do with 90% of academia math. Also, if you specialized and are interested in a certain field of math, the barrier to switch to another field is enormous. On the other hand, 90% of the applied mathematics done in industry is much, much simpler, and very often a trivialized version of what you do in academia, so one could imagine switching to be possible, but what's the joy of doing that?
(I have a MS in Math, my field was algebraic geometry. I considered a career in math research, but the industry was paying too well for it to be a reasonable)
In fact, in academia, in some fields (and mathematics is certainly among them), in some sense only "the best" contribute to pushing the field forward by working on the cutting edge.
And I think the author's point was that focusing only on that is an overly narrow view, and one can engage in maths in a recreational, joyful way, without aspiring to an academic career in it.
Yeah, I'd argue a very un-Good Will Hunting thing: if you can be convinced you suck at math, you probably should not pursue a career in academia in math.
However... being a math major is really different from a career in academia. People really should be encouraged to major in math.
I would have to agree with this, but I disagree with out math is taught by most schools.
Somewhere in this thread there is a link to Francis's lectures and they are wonderful. The method/enthusiasm be has for teaching is wonderful and makes all the difference to students.
Rather than going through online courses and paying for them (many university classes are mostly online math problems, and proctored during a specific time.)
EDIT: The way math is currently taught really forces people to dislike math and think they suck at it. (in general)
"EDIT: The way math is currently taught really forces people to dislike math and think they suck at it. (in general)"
Starting in primary school with basic arithmetik.
(I help a young boy with his homework)
They don't really teach them Math, how to solve thing, they tech them to memorize different algorithms, which they can handle after a while, but don't know what they are doing at all ...
So the best mathematicians there are not the ones who can think best, but who can memorize and follow orders the best.
I think this is true in some cases and not true in others.
I met a woman from South Korea who was taught all drill and no concepts. She was solving differential equations in high school but had no intuitive concept of derivative and integral or what they were used for. In fact, she had no interest in math. She was just a diligent student focused on getting into the top university. She was obviously trained the way you describe.
I have also personally, in the United States, been in math classes where many students suffered because they couldn't string together correct calculations consistently enough to validate and reward their high-level understanding. They learned a lot of the words and pictures, could explain what an integral was for, and could listen to a lecture and feel like they got it, but if you asked them to apply what they knew to a real problem, they responded with a kind of rueful helplessness. To them, mathematics was like magic in the Harry Potter universe: anybody could explain it, but it worked for some people and not for others, for reasons that seemed to them to be innate.
Each system stressed one aspect at the expense of the other, and in each system, there were many students who picked up both, but also many students who only learned the part that was stressed by their teachers. It was certainly the case in my classes that a student who only learned the concepts, without the mechanics, was unlikely to progress much farther in the math curriculum.
A balanced method treats the two aspects as complementary, each enabling the other. Treating one as the hero and the other as the villain might make sense locally as a response to a warped system, but it can easily become a warped approach in itself.
Of course, most students do just memorize equations and follow orders. But in my experience the most successful students are always the ones who understand the material on a fundamental level and memorize very little (on some level, memorization is all but required by even the best).
The successful students do that in spite of the bad teaching. In my view, a lot more students would both enjoy and do better at maths with more appropriate teaching.
> They don't really teach them Math, how to solve thing, they tech them to memorize different algorithms, which they can handle after a while, but don't know what they are doing at all ...
Common Core in the US is supposed to solve this. I've seen some common core math homework for myself and despite the odd outrage most people have towards it, I think both the idea and execution are at least decent, and an improvement.
I can see how that can work as well, but I would argue having a deeper understanding of the problem (whatever it may be) will benefit any mathematician/student in the long run.
It works, if you don't want to think and just get results. But today there are calculators for that, so I'd like it if they teach them how to think, not to follow.
That's not a counterpoint, that is exactly the point. Maybe it becomes clearer if you read the speech. Math is hard, but it's being made much, much harder than it has to be.
What's the alternative? I mean that seriously, not sarcastically or in a defeatist way; try to explore what the alternative might actually be.
On the one hand, academia has become rather harsh and intimidating and there is room for all kinds of improvement.
On the other hand... there is no world where whoever just wants to study math can just go study whatever they want for as long as they want, regardless of how well they do it. Of course, when I spell it out, that probably seems obvious, but I suspect this may be the unexamined assumption in a lot of people's heads.
Would you say the same about exercising and sport?
"There is really no point in doing any exercise unless you're aiming for Olympics level performance. Yes, it's somewhat harsh and intimidating. But there is no world where whoever just wants to can just go do sports and exercise what they want as long as they want, regardless of how well they do it."
I think that was the point of the article - doing maths as a way of life (like exercise), not just for your career with the aspiration of being the best.
There isn't a world where you can just go do sports and exercise as long as you want, regardless of how well you do it. Eventually you're going to have to eat something.
I'm speaking in a context where we're talking about academic life and academic life being harder than it needs to be. I think that's the only way to read this context, because nobody is making self-study "harder than it needs to be", so reading this thread as being about "life in general" doesn't make any sense to me.
When you have a decent work-life balance, you can certainly engage in sport and exercise, even if it is not jour job.
I think the article is precisely not about professional mathematicians in academia. To quote: "If mathematics is a medium for human flourishing, it stands to reason that everyone should have a chance to participate in it." (my emphasis).
I solve Project Euler problems for fun. The article presents an argument that more people should do something similar, and more mathematicians and maths teachers should support it.
"there is no world" - wasn't that what University was like before the 80s and still is in many other countries? minimal fees and some casual work and you can stay for as long as you can stand the conditions of the average student flat ;-) I think you are conflating the reality of your current culture with what was/is in other cultures.
If what you describe was even remotely the case, which I don't know, then there was some other filter in place preventing "everybody" from doing it. I can tell, because not "everybody" did.
Plus I suspect you missed my "regardless of how well they do it" clause. I doubt that you can flunk every math class over and over and just keep attending. Loopholes that big get noticed, exploited, and closed very quickly.
In German universities in the 80's and 90's you had people studying for decades. It wasn't a "loophole", either, it was a feature. Universities were there to teach, basically everyone could go and study.
In recent times, the pressure to finish or drop out has increased, and studies have become more rigorously structured (or infantilised...), particularly with the introduction of the Bachelor/Master system.
There was a very strong filter. It used to be very difficult to get into university before they expanded tremendously from the 70s to now. You needed excellent grades. Now good grades are enough to get a spot somewhere.
there is no world where whoever just wants to study math can just go study whatever they want for as long as they want, regardless of how well they do it.
Certainly in Sweden, once you're in, you're basically free to hang around at University and study for as long as you want. It's not like they're going to kick you off campus.
An alternative would be a system where publishing isn't as important as obtaining results. Let small journals take care of everyday research and let only the big works end up in something that really define someone's carreer. There would be a lot less people with a "valuable" publication, but we do have other ways to judge people and assign spots as researcher. The problem is the way jobs in academia are assigned: the more you published, the more you get paid. This shifts the attention from knowledge to publishing. Just value people with the usual attitudinal tests unless thay discovered something truly remarkable.
Wow, I always shied away from academics for similar reasons you mentioned, but I just realized that my main hobby, game development, parallels what you say a lot (and creative hobbies in general, including writing).
"You should expect to invest years of your life into making progress on some meaningful game with little encouragement in the meantime...sometimes even after you release a game that took a lot out of you, it may take many years before people appreciate, let alone play, your game, and the vast majority of games aren't going to be played." "To compensate for this you basically play the popularity game and give as many interviews and talk up your game when you meet people, so there's a lot of salesmanship involved there as well."
"The career path is so twisted you have to either be insanely good or just hate yourself enough to sacrifice your best years working essentially in the metaphorical darkness well outside the spotlight and most likely alone and poorly paid." <- My experience working in the game industry, except I wasn't completely alone, I did have coworkers. However I did feel like I never had time for a relationship.
We need to do better at cultivating interest in math, as well as other "hard" fields, among people who are normally excluded from the system.
Su isn't as concerned with the amount of full-time mathematicians doing research work as he is the people who drop out far before that becomes an option. We should teach it better in high schools, and encourage more math undergraduates.
Su isn't saying math is a great field free of problems, but he's arguing we should focus more on getting people further along that path, regardless of the stage where they drop out. I also think the problem you're referring to exists in most academic fields, and that the solution (if there is one) will probably have to be more general than just relate to math.
The article is about math, not necessarily academic math. And it was far from "overly exuberant". Jesus. I have no degree, and do math all the time, precisely for the reasons listed in the article. There endless comments shitting on academic life are obnoxious and tedious, they are also becoming something of a plauge here.
Since I made this comment, I also had the chance to listen to his lecture in full. I couldn't agree more with him. What I wrote here was based on my understanding of what he meant by "doing mathematics", which as other commenters also pointed out earlier did not have anything to do with a career in math. I stand corrected.
Cool! Thanks for taking the time to circle back and add this. I was also responding to the wave of "academia sucks!!!", which, while perhaps true (for some) is getting really old to read in threads here. Appreciate you taking the time to add this comment.
Do you think it's the culture of science or the inherent uncertainty?
From my own experience I think it's the latter. Jumping off the edge of human knowledge and hoping there was a result to catch my fall was very mentally taxing. Not many other careers force you to grabble with the uncertainty of creating knowledge, and I think it takes a toll.
This has been my experience with trying to get a foothold in the technical civics communities, so I don't think it's either. I've found that the bar gets set pretty high by folks who are willing to speak loudly about their accomplishments in a ways that exaggerate the actual accomplishments. If you're not really good sales person towards your own work, you find yourself working significantly harder than what's probably needed since fewer people are willing to work with someone who hasn't had any accomplishments. It really sucks that ego and salesmanship gets in the way of trying to do civics work.
How many people can live like that in any practical sense? Everything I've ever read about motivation, learning, flow, and mastery holds quick and direct feedback as the most central necessity. Applied to software development, hence we prefer fast compilers over slow, unit tests over integration, working on a desktop over an SSH connection halfway around the world, frequent releases, etc.
It sounds like you're not being sarcastic. So let me say that when you're working on something that almost surely has zero practical impact over the course of your lifetime, and the only possible benefit to society is the advancement of the state of our knowledge, a lack of enthusiasm from your peers can be gut wrenching no matter how mentally resilient you think you are. No one does math solely for appreciation from people they admire and respect, but the lack of it, even in the short term, can be very hard on a person.
Scientists actually spending more of their time reading others' work, communicating encouragingly and giving constructive criticism (anyone in academia has read more than a few hate filled reviews of their work), better pay (a postdoc at Oxford nets you a monthly salary after taxes of about 2.4k pounds), less focus on numbers and more on quality (a big reason why people don't read papers is because they're missing out on little by not doing so i.e., much of the work is incremental, unsurprising or just badly posed or ill motivated. instead focus must be given to quality and this must come from job hiring decisions which should reflect that science is not a numbers game, but the sad sad truth is it so very is). About the encouragement part, I felt let down many times by senior scientists by their uncaring and viciously competitive nature. All this does not need to be a part of academia. I think greater science can emerge after these structural problems are addressed. But currently the only people who rise to the top are the genuinely good ones and the ones who know how to game the numbers system. Idealists who come for the beauty of math often find themselves woefully unprepared to play all these other games in addition to doing good math.
I took this to be a general call to learn and practice mathematics, and not as a summons to a career in academia, so perhaps I'm seeing it a bit differently.
I have to say as somebody who through attrition always feared math in university that it is meaningful to think about reasons why people fear embracing certain types of knowledge. It's only after the fact that I realize how useful and even how romantic applications of mathematics might be.
I might never be a starry-eyed or an often-drunk academic, but I've grown to really see my initial lack of mathematical learning to be an immensely high opportunity cost. So I can relate to anybody who thinks of structural reasons people might shy away from math and indeed all forms of intimidating knowledge.
>how romantic applications of mathematics might be.
like, in the dating market or do you mean a philosophical potential? The philosophical aspect is huge, eg. to tell once meant counting, or logic and language having a common Greek root.
The problem with this theory, is that it completely ignores that a lot of people good at math have poor people skills. And you need people to be happy and improve your life opportunities.
So yeah, math is beautiful, and if you like it, go for it.
But if you look for a skill to acquire or practice, your taste no withstanding, this may not be the best investment. Sport, social skills, languages, time management, self introspection and cooking are examples of things that usually pay off better than math in your life. It brings more people, opportunities, health, money, etc.
Again, not saying math is not a good thing to practice. We need math as a specie. And an individual may need it for his or her happiness. But as a strategy I don't think so.
You could likely be linking an association with a trend. Maybe society treats people who have the personality requisite to study mathematics poorly. Thus, over time, both the individual and society 'negatively associate'.
That was my experience, anyway, and I know it has been a historical trend. Some of the best mathematicians were bullied by their cultures, isolated, and thus had plenty of time to work on mathematics.
But I've been a young nerd with nerd friends, then working with nerds.
By some twisted fate, life forced me to improve my people skill a lot. It was painful, I didn't ask for it, but now I'm glad it happened.
Now I still have a big nerd entourage, because I like to be surrounded by people smarter than me. But I can't help but notice all the things they do that drive others away. Things I used to do.
And this things are linked to they nerdy skills.
E.G:
- they don't play artificial social games
- they don't try to dress up
- they are straight forward and honest
And they are because it would not make sense for them to be otherwise. It would not be logical. I would be stupid. A waste. Useless.
So they don't do it.
They also usually have a strong ability to feel, connect with people. But they also use they intellect to deal with those instead of knowing how to manage their emotions. But using a screwdriver to nail something is inefficient. And people are so overdoing it. Touching, making noise, taking attention, expressing themself in overwhelming ways. This is utter nonsense to them. It takes a toll.
So they take a distance.
And the world is always agitated, filled with unimportant things, requiring to switch context, to take stances on unnecessary things.
So they stay in their head.
And people are petty. Why would anyone not do what's best if they know it is ? Why would not somebody work for the group ? Why would somebody purposely destroy something ? Pick up on someone ? This is mean, it makes no sense.
And why would they not use proper words. Understand this efficient sentence ? Get the reference to this book / movie that is so good ? Talk about those stuff, sport / terrible tv show / cars that are such bullshit. Ignore scientific evidences to run their life ?
So they separate themself from part of the population.
These traits, I see them over, and over. But it's what allow them to analyze. It's what allow them to concentrate. To remove clutter. To categorize. To extract. To model systems. To abstract. To understand. To solve problems. To bring solution. To makes things better.
These traits you describe, you say they are a side effect of good reasoning skills. I agree that people who think more methodically are more susceptible to these traits, however, I think they are actually characteristic of a certain immaturity in one's reasoning abilities.
To ask : "Why would anyone not do what's best if they know it is?", and to have no clue of an answer, would imply a lack of reasoning ability, rather than a surplus of it. Likewise for the other questions you suggest.
I think this way of thinking is very natural to a young mind who finds the world overly complicated and is attracted to the logical nature of subjects like maths. It is a sort of escape, to mentally write off everything that is not clean and simple as "illogical".
Clinging to that world view for long time would require serious cognitive dissonance though. It is a crutch that those with a 'problem solving' mindset hopefully use for a while before seeing things with a more subtlety.
I don't claim to know what reasoning maturity looks like, but I believe embracing your humanity and own fallibility, and understanding the motivations of others (especially those you dislike / disagree with) are not trivial elements of it.
I do feel like we are on a similar page though, since you are describing the traits of others, not your own, and you say you are happy to have developed your people skills. Still, I worry that explaining these traits away as the result of above average intellect is some sort of enabling. It's the narrative they need to stay in place so they don't have to change. IQ becomes a justification for being detached and uncaring.
> IQ becomes a justification for being detached and uncaring.
I see your point.
But I think some things are like a color you never saw and can't understand until you see it for the first time. In a sense, it's lack of maturity because it's a lack of experience.
You can wait for the experience to reach them, or you can offer them to practice it. The second way is quicker and more reliable.
I agree. I don't think you can get past that stage without a few 'aha' moments that make you rethink things.
The mindset might be a bit of a trap because it can make you more isolated, which could then make those 'aha' moments less likely. In that position, you're right, waiting for it to fix itself won't be very effective so a more proactive approach sounds better.
While it's true the current culture is particularly hostile to math incline people, I must also say that, by nature, people good at solving problem with perfect solutions are not good at integrating in an imperfect society.
An individual is but one individual. Societies ills should not be dismissed nor diminished because of an individual or even group. Society affects everyone in vastly larger ways.
I'm thinking of it as similar to a companies products. A company will refund you if you bring them evidence that they sold you a defective product or service (in a good state of the world). That individual bad product hurts their bottom line, but typically in a small enough way to be mostly ignorable. Now when the company has to issue a multimillion dollar recall some internal process or procedure to the company needs to change. The general operating parameters of the company affect its overall "health" much more than a one off "bad product". In this analogy, societies are the company and it's individuals are the products.
In my experience, people are all equally bad at math. There are just people who have spent longer with the problem(s)/domain(s) and have developed a skill set around it. They start from the same beginnings, though. If you take them out of their usual 'watering hole' they go straight back to beginner status (and of course the degree of it varies by how far it is from their usual routine).
Yes, but some personality traits will make you more apt and willing to keep working on it.
There is a limit in the skills you can acquire. If you spend time and energy on flirting, learning to dress up, building social network, you will have less time to learn maths.
Because of course you'll have sport, music, games, books, movies, family and other study topics taking time as well.
Now maths incline kids, given the choice, will usually choose working on math than going to 10 groups of people to chit chat about superficial topics just to stay in the network.
See my other comment where I explain in details what I mean.
I met some people that are both good at maths and with people skills. They are, however, adults. They started as nerds, and worked their people skills on the way up.
To be fair, I had a strange life. I changed school 11 times, and rebuilt my social circle every time. My father also worked for an airline company, so I traveled the world most of my life.
Because I was a geek, I was mostly surrounded by geeks. It gave me a huge and diverse sample to observe. Granted, it's not something you can consider scientific, since I'm the common ground and so the bias in all those observations.
You won't achieve that by encouraging people at doing math, it's not attractive enough. Just like it's hard to sell vegetable to somebody eating chocolate. And I LOOOOVE vegetables.
You can, however, achieve this by encouraging nerds to get people skills. Double benefit :
- they will have an easier life
- nerds will be less stigmatized, linked topics will so be more appealing and math more successful
I agree though that putting more kids in contact with math sooner will help a lot. A lot of kids like maths if it's shown to them with passion before they are exposed to the social stereotype or nerdiness.
Well, mathematics is hard. It takes a lot of effort to really learn it, a lot of dedication and (self-)motivation. Not everyone can do it, and, frankly, very few people - even among those who have spent many years studying mathematics - could say that it is the best way to live your life.
I profoundly disagree. I believe that mathematics can be learned and appreciated by many, because mathematics is a language for articulating inner urges and perceptions that are common to a wide range of creatures, not only humans. This idea that mathematics is only for the few is self fulfilling, and perpetuated by narrow ideas of what it constitutes.
I agree. I've noticed this issue where your anticipated difficulty of something ends up determining your experience of its difficulty. With math—most of the time—there's nothing intrinsic in what you're learning that would make it more difficult than say, learning the rules to a complex board game, or the structure of one's government—and yet, people go into learning math with knowledge of its reputation as only being suited to certain special kinds of minds. If you suspect you're one of the chosen you can go into it with relish; if not, you respectfully put in your time, knowing you're in someone else's territory, and sort of meekly do what you can.
I think most of the difficulty is 1) determined by your expectation of it 2) initial foreignness and the invisibility of the gap between the concepts you need to understand something, and the ones you already have (i.e. when an outsider just looks at some mathematical statement they have no comprehension of, it's not clear that there are a few layers of concepts underlying it which could be smoothly traversed, as long as one puts the time into it and is given some direction.)
> I've noticed this issue where your anticipated difficulty of something ends up determining your experience of its difficulty.
That assessment didn't come from nowhere.
Given that there seem to be aptitude differences for most physical things, I would find it hard to believe that there are no aptitude differences for most mental things, especially for something so unnatural.
Not to mention that all the things you speak of are very real difficulties to many people that they're not going to get any visible pathway through.
> I believe that mathematics can be learned and appreciated by many, because mathematics is a language for articulating inner urges and perceptions that are common to a wide range of creatures, not only humans
I'm not sure where you're getting the 'wide range of creatures, not humans' part, or even why that matters, but I agree, mathematics is a formalization/projection of our own thinking process, which pretty much every human has.
I agree. I think that mathematics is being teached the wrong way. Especially in the USA, the state of math education is absolutely terrible.
I suspect a more visual approach to be more useful than the classical textbook approach. It is true that a lot of textbooks do use pictures, but video's would help a lot, I think.
There is a wonderful book, "Visual Group Theory", that demonstrates this approach [1].
However, there are people who learn better algebraically than visually. So, combining different approaches would probably be optimal -- the problem currently is that often a very dry (and anti-historical) endless litany of definition, theorem, proof is taught.
There is a huge difference between the pursuit of new mathematics and the appreciation and understanding of existing work. I think OP is talking about the former and it certainly isn't for everyone, at least not the way it is currently implemented. To understand this you need to look at the career track of newly minted math PhDs and what one needs to do to secure tenure.
I do agree that math can and should be widely appreciated.
> To understand this you need to look at the career track of newly minted math PhDs and what one needs to do to secure tenure.
You can pursue new mathematics without having a PhD. I think the most difficult/frustrating part about academic/institutional mathematics is the politics and bureaucratic bullshit.
It's the fallacy that has led to the extreme oversupply of post-graduate academic job seekers: far too many people are in college, getting advice from professors who think, "well, it worked for me, you should do it too!"
I agree, I watched his lectures when trying to teach myself real analysis. He also has a website with lots of useful info including links to the videos organized by lecture number/subject and some other interesting links
I literally start to love mathematics with him. Before watching his lectures I always afraid of mathematics, but watching his Real Analysis class, developed a new sense in me about mathematics. From that point I always liked mathematics.
Francis Su is a great person. I've had the privilege of meeting him and having a couple of quality conversations with him. He is (or at least used to be) a regular part of the faculty at the MathPath camp, a summer camp for 11-14 year old kids. My daughter absolutely loved that camp, and attended for three years. Su taught some very fun sessions there that my daughter (now college freshman) still talks about. (BTW I highly recommend the camp.)
I found it amusing that the article referred to Harvey Mudd as a "liberal arts" college. I think of it as a butt-kicking engineering school.
Thanks for the link! I'm going to try to set aside some time to watch these lectures in the future. I've always wanted to learn more math in my spare time, but I've been unsure how to approach it outside of a classroom.
It seems like a lot of the benefits mentioned could arise from many other activities with similar likelihood. I think a lot of it has to do with developing expertise at something you believe is important, and which you are (or can become) comfortable doing.
Totally agree that for people working in abstract technical areas (e.g. software architecture, philosophy, inventing things), however, mathematics has a special sort of value over other subjects. It deals in these super distilled concepts which have very general applicability; so, the concepts you learn end up expanding this pool you can draw from for coming up with new, related ideas, in a wide range of fields.
It's also important to learn it as a sort of literacy, to widen the range of technical material you can read.
Well maths always sounds good with all its allure of "purity" and "intellect". However I think one should remember that maths is basically some very-high level abstractions on our analog, chaotic world. Just like rigid linguistics/grammatical rules fail miserably in representing actual human languages, abstract maths probably also cannot claim to represent the real essence of the world that well. Even a statistical/neural network-style approach does it so much better. So yeah, if you're so into constructing and deconstructing abstruse abstractions, maybe do maths very seriously. But does it really represent the "truth" of life? I think that might be a bit dubious.
I find the headline and the article in the OP to be a bit misleading. This is not about the general public, or the relationship of individuals to mathematics. It's not a "hey, you should do more maths!" type of motivational speech.
The whole speech is addressed to the leaders of the math community, i. e. the teachers. It's a well-crafted plea for introspection, to find the humanity that was somehow lost along the way.
Give anyone (not just inmates) a computer and they are much more likely to become good Candy Crush players, Facebook users, and Buzzfeed readers than good programmers.
Sure, just like give anyone access to a library and they are much more likely to read magazines and romance novels than 19th century French literature.
That's missing the point though. The point is access and opportunity. Even if only 5% become programmers, it's a net win overall (and that's not even counting the people who might play Candy Crush but still use the computer to better themselves in other ways - learning, filing taxes, etc.).
I would assume they would being writing. That would likely be the first step most would take. After that their abilities in mathematics, or just a general interest in problem solving would take them pretty far (barring games and internet)
... and a device that lets them play a handful of games like tetris, but only 1 minute/hour. They'll get to work re-implementing their favorites (without the time limit) on the computer before long.
While I can not claim to have made any imporant progress on the sorts of toy problems I like to work on (e.g. counting/bounding the number of strings two regular expressions both accept, counting the number of ways to represent integers using polynomials), I find the mere process of working on them, inventing new notation to succinctly express ideas and read relevant (and completely irrelevant) mathematical textbooks and articles a way to keep an "active brain" but relax: probably because nobody expects me to produce anything and my career doesn't depend on it.
Elementary math skills, which many of us reading and writing on Hacker News know, would change the lives of many millions of people in ways we take for granted.
The level of math amenable to apps isn't really what Su is talking about.
I would suggest learning proofs, and maybe pick up some Art of Problem Solving books. Or perhaps working through the foundational curriculum of any decent math program (e.g., algebra, analysis, topology, number theory, etc.).
Khan Academy can get you pretty far, if you like going through the remedial stuff then you can even take the quizzes to get "Mastery" in various subjects like Algebra, Trig, I think Calculus too. But I'd love something else that is more adaptive and gives you exercises and education as you go. As a hobbyist one of the hardest part is figuring out where to jump in so you're neither bored nor overwhelmed.
I was recently explaining to a group the radiance of learning math for many years. It is really hard to convey the beauty that you see in things after years of studying math/logic. That amazing feeling through your mind when you solve something new. I can sometimes only best describe it as a lens that allows you to sense a previously imperceivable part of the universe and self. I can't even imagine walking around without this lens. Perhaps that sounds a bit hyperbolic or mystical, but this is my feeling.
Based on all the supposed benefits of doing math, it sounds like people would be better off studying philosophy. Same benefits but a much broader appeal, and far more applicable to most people's everyday lives.
When I finally "discovered" philosophy in college, I was angry that we hadn't been exposed to it at all in middle or high school. Instead, I'd been forced to waste years on things like math, biology, etc. that I had no interest in and no use for. Our history / social studies classes would be greatly improved if they incorporated more philosophy.
Real philosophy can't be limited to some set of tools or into some set of books and approaches that fits into curriculum. Philosophy is completely open ended pursuit.
I firmly believe that philosophy without mathematically oriented thinking at the core is limiting. Teaching and studying of formal logic has been traditionally part of philosophy. In practice there is no real border between philosophy and formal sciences (logic, mathematics, statistics and theoretical computer science).
Consider the problem Kant faced in Prolegomena to Any Future Metaphysics, §13. Two objects that are intrinsically alike must be interchangeable. But there are objects that are intrinsically alike and you can't exchange them.
Kant was incredible philosopher but he was thrown off by looking at his hands. Left hand and right hand seem to be intrinsically same, but you can't replace one with another. Kant concluded that things like chirality and mirror images could not be understood with intellect and reasoning using concepts. Time and space were part of sense intuition. Mathematics would disagree. Spatial intuition is not fundamental building block for reasoning about time and space. Algebra, geometry and topology are.
Many modern day problems facing humanity can be understood only as systems thinking using, probability, mechanism design, games, incentives, equilibrium, trade-off, hysteresis, etc. But many schools of philosopy still try to use tools and concepts that can't describe the system.
edit: Several important contemporary philosophers and schools of philosophy are not are in fact limiting themselves. Saul Kripke for example.
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I think philosophy should be part of curriculum in school. But it should be live philosophy centered in problem solving and rational thought. The traditional history oriented curriculum should be part of history. History of science, technology, philosophy, economics and ideas in general is more important than the narrative trough kings and power cliques.
I think this is an excellent description of where philosophy has gone wrong for me personally. I took a class in high school and found the study of what amounts to history to be rather tedious and fruitless. A close friend was a philosophy major in college and understood lots of abstractions that came from the field, but it would seem that none of them were applicable to everyday life. They were not useful or practical in any fashion.
It's very much like learning the history of mathematics without being able to e.g. solve an integral, or model a problem in the real world and make meaningful observations about the model and its relation to the subject. History is fine if it's what you're into, but the other is not taught in any widely disseminated fashion and I believe people and society are deprived as a result.
There are entire sub-disciplines of philosophy that are useful and practical for everyday decision making. Here I would disagree that you need a formal background in mathematics. Game theory, statistics and the like are certainly valuable, but they are still more a part of the engineer's or mathematician's toolkit and will be glossed over due to their rigorous technical underpinnings that are simply out of reach for many individuals. Rhetoric, critical thinking, stoicism, ethics, and the like are pretty approachable topics that have somehow been elided from the educational system in any established fashion in the US - and they used to be at the core of a liberal arts education. Great religious thinkers, politicians, and intellectuals have left a legacy of work that speaks across time and space to a modern reader.
There is an analytical strain, of what I consider a better class of philosophy - see https://en.wikipedia.org/wiki/Logical_positivism , just it tends to be in a minority among university courses. I had to seek it out.
A lot of 'continental' philosophy seems to be a mix of historical analysis, and cult-of-personality/lit-crit...
In PTAFM 13, Kant is arguing that neither space nor time are intrinsic properties of things in themselves, and basically making the point that properties like position and congruence are relational, not intrinsic. Spatial relations fit with the intuition of space, which is the form of external experience (and one should think of this as the abstract form of any space whatsoever - people tend to think that Kant is undermined by the development of non-euclidean geometries, but I think one can push the abstraction so it fits equally well). Experience of- and reasoning about spatial objects involves both intuition that corresponds with the forms of space and time (which applies to any experience whatsoever, outer or inner), and the operation of the understanding through concepts. Now, reasoning from principles which might apply to spatial objects (in the case of algebra, geometry and topology) can go through concepts alone as long as it is logical. But those principles would be meaningless for us if we didn't have experience of any spatial objects whatsoever (if we didn't have the pure intuition of space).
"Thoughts without content are empty, intuitions without concepts are blind. The understanding can intuit nothing, the senses can think nothing. Only through their unison can knowledge arise." (KrV A51).
This highlights why math generally trumps philosophy. Take any math that is similarly well known to Kant. You don't get two educated people having diverging opinions on a solution if they start with the same assumptions. Usually someone concedes; e.g. because the other side shows that the losing position implies something that everyone accepts is impossible.
In philosophy you get this all the time because people refuse to agree on definitions like truth, consciousness, goodness, evidence etc. So debates generally involve people talking past each other until someone gets bored. If we're lucky spectators will judge by popular acclamation which side won or lost. Participants rarely concede their position.
> You don't get two educated people having diverging opinions on a solution if they start with the same assumptions.
No, in this case he was mistaken in some basic assumptions about what Kant is doing in that text. The paragraph he mentioned starts with: "Those who cannot yet rid themselves of the notion that space and time are actual qualities inherent in things in themselves, may exercise their acumen on the following paradox." So in this case it isn't even the case that he is proposing something about Kant one can reasonably disagree on (like my suggestion that Kant is not undermined by the development of non-euclidean geometry, something which I would be willing to concede given evidence against it).
On the other hand, in my experience, actual debates on philosophy usually revolve around conceding some initial assumptions and then debating on what follows from them. Of course, one can always move from the internal questions about what follows from those assumptions to external questions about what follows if we rejected those assumptions, but this move is usually well motivated by internal conflicts. One might, for example, find some dilemma for which no option is acceptable, and thus be forced to retreat to discussing the assumptions. And it is not a given from the outset whether no such conflicts will arise.
The problem I got with philosophy is that it's supposed to be applied knowledge. But it's always consumed as an intellectual one.
As a result, the smartest philosophy lovers that I know are incredibly unhappy. They know so many things. But knowing something doesn't mean you are able to do anything about it.
Some becomes cynical. Other become depressed. Other procrastinate to hell.
But only those who actually take the knowledge and try to act on it, improving themself in the process, ends up happy. And once they get there, they usually don't quote much philosophy anymore, except for humorous purpose or make someone feel better.
Bottom line: it's interesting to look for the meaning of life. But it's necessary to actually stop and live your life unless you want to have the joy of Sartre, the energy of Rilke, the sense of purpose of Kant and drive of Schopenhauer. Hint: you really don't.
The problem I got with philosophy is that it's supposed to be applied knowledge.
Can you elaborate on that? On why it's supposed to be applied knowledge I mean?
To me philosophy is one of the most abstract forms of thought available, and to me abstract is the antithesis of "applied". Maybe the furthest I would go is to state that philosophy is knowledge applied recursively on itself. But that already seems an abstract formulation to me.
Because eventually, you can always go deeper and more abstract. The more you do philosophy, the less you get answers, the more you get questions, and they become less and less practical. At best it becomes absurd or pointless, at worst it becomes depressing.
However, if you use your knowledge of philosophy to actually do something in your life instead of trying to think your way into the abyss of infinite dissection, it becomes a powerful tool.
This is the main difference between very old philosophers (Seneca, Gautama, Epicurus...) and more recent ones (Kant, Sartre, Schopenhauer...). The first ones try actively to make something out of their thinking. They derive a way of life out of it. The second ones try to understand more deeply, categorize more things, raise more difficult questions.
In the end, by reading the active ones, you can feel the joy, the sense of purpose and solutions arising. If you read the other ones, you can almost picture them trying to kill themself.
I disagree. I find math much more interesting. Philosophy is vague. It's all guesses and interpretation, with the exception perhaps of pure logic. I like the purity of math. It's all a matter of preference.
The concepts you deal with in philosophy are vague, the challenge is to provide very precise analysis of these vague (fuzzy?) concepts.
But yeah I don't think one is better than the other. Each provide value.
I would actually claim that to live your best life, be an engineer. They seem to me at least to be living in that sweet-spot between the abstract, creative and the concrete.
This has a 99% probability of being absolute nonsense. Looks more like 30 years of anecdotal evidence and hearsay. Try find a credible source with actual numbers.
I'd love to know where that 99% comes from. (Or maybe I wouldn't.)
I'd say there's far more than anecdotal evidence, but not, obviously, not enough to draw necessarily firm conclusions. So I would say there's enough question to sustain further scientific inquiry. That, interestingly enough, puts us in a position at odds with one of a common traits noted among engineers, in psychological terms: 'need for closure'; which would put the 'engineering mindset' in conflict with the 'scientific mindset', due to the open-ended nature of the science. (Of course, this is only my own speculation.) Which is why we get things like this: http://cosmicfingerprints.com/ee/
I pulled the 99% out of thin air, I just wanted to emphasize how small I estimate the chances of this being true. I found one analysis [1] that found that creationist or only half as likely to be engineers as the general population, admittedly only using data from a preexisting survey as proxy for the question at hand.
Maybe it is a naive view on my side but I expect that additional education will make it less likely that one believes religious claims to be true in general and this extends to becoming an engineer and being a creationist.
But even if there was indeed a correlation of the form claimed by the Salem hypothesis, I would naturally want to look for traits that make it more likely for one to become an engineer and a creationist, not for something that causes engineers to become creationists.
You did not explicitly spell it out this way and I am inclined to think you do not think this causation exists, but your response to a comment suggesting that it might be a good choice to become an engineer at least allows the interpretation that becoming an engineer causes becoming a creationist.
And I obviously consider the idea of a causal relationship between being an engineer and being a creationist even more unlikely than that of certain traits increasing the likelihood of becoming an engineer as well as a creationist.
Not that it is unlikely in the general case that learning about X makes one more likely to also believe Y, that is actually certainly pretty common, but in the concrete case I am really unable to see which things one learns when becoming an engineer are suitable to turn one into a creationist.
Finally I am not sure what you wanted the express with the Evolution 2.0 article, but at least in the linked article the reasoning is heavily flawed.
Most philosophers are not very careful thinkers. There is a lot of circular thinking a lot of apriori assumptions and so one. Mostly the designers who are worth listening to IMO are those who broke down previously held illusions.
I always saw philosophy and maths as the same. Philosophy is what math and logic especially looks like when you're using words instead of leibnitz notation and other modern notations.
I mean, howdo you explain modus ponens without the notation? You say that not A and not B is the same as not A or B. And that's a poor transpilation from math notation to words. What if you never saw the math notation and had to use regular words?
I do as well, to some extent. However, let us never lose sight of the fact that any foundational axioms (e.g., ZFC) are decided upon on a purely philosophical basis. It is interesting to see such philosophical consideration around the "behavior" of truths when reading about Frege's early developments in propositional logic.
Like in math, it is humbling to revisit philosophy from a historical perspective and to be reminded of the cumulative impact of simple ideas (to us) that were being considered 2000+ years ago.
I agree, but there is a lot of value in approaching Philosophy from a mathematical context (both as a student and as a philosopher- c.f. Russel, Wittgenstein, et al had backgrounds in mathematics and left a huge mark on the subject).
I was a math major for much of my undergraduate career, but by my junior year I realized that I was actually only interested in the philosophy behind it. In fact, I had little interest in my freshman Philosophy 101 course, though after studying math, I realized that I would have loved to study philosophy more. As a freshman, I was enamored by math as it was this ivory tower of abstract truth.
I loved learning about what math could say and how the notation worked and the notion of formal proof etc. but slogging through proofs was abhorrent after awhile and I dropped the program. I had come to understand that math was actually just the same sort of discourse as "soft" philosophy, just formalized. Learning about counterexamples to math as a "closed system of reasoning"[0] (Godel's theorems, constructive mathematics, etc.) simultaneously ruined my perception of the nobility of math and spurred my interest in philosophy (specifically analytic philosophy and philosophy of language) and cognitive science. That's where the real "unsolvable" or "interesting" problems lay (e.g. the mind-body problem).
I would highly recommend anyone who is intellectually curious to learn and understand formal mathematics for the purposes of understand philosophy, but also to recognize when to quit (if ever!).
[0] - I may be butchering these terms, but I hope the meaning is clear
> simultaneously ruined my perception of the nobility of math and spurred my interest in philosophy
Can you explain this? Math isn't perfect, so might as well abandon all formality? This feels like "Science doesn't have all the answers, so maybe I can find them in the bible?"
I'm not sure how constructivism hinders mathematics, or Godels in the long run.
The mind-body problem is unsolvable because it rests upon vague or false assumptions; It is philosophical nonsense : "communicating badly and then acting smug when you're misunderstood is not cleverness." https://xkcd.com/169/
You could make these type of arguments over and over. How about doing multiple? Math, philosophy, music, dance, cook, workout, just do the things that you enjoy, and do meaningful things. Dance to your own damn beats whatever it is that plays in your head and tickles your fancy. last year, math it was and for about 4 months straight, I studied maths every morning, then I left it and started tooling around with the piano, then I started coding, now I'm coding and tooling with the piano again. I do have some philosophy books waiting to be read. I'm living my best life. Whilst you might have no interest in math, biology or other subjects. Some people do. I do have use for math and interest. I wish I had strong biology and chemistry background, bio engineering interests me, but I can only watch from the outside.
Math classes ramp up in a gradual way that develops critical thinking and intuition in a very small sandbox, where it's easier to appreciate the results. Thinking in, say, just the x-y plane makes it far easier to isolate relationships than philosophy, which tackles much bigger problems and gets you into controversial problems almost immediately, with no clear answers. The basic math framework is rigid and precise, whereas philosophy only gets that rigidity at much higher levels where, guess what, it starts to merge with the field of mathematical logic.
Of course, a one size approach won't fit everybody.
To me the key point is "no clear answers" in philosophy.
In math you learn to follow a series of developments, building structure that leads to statements that are true. These are the foundation for more bricks in the wall and even bigger constructions. e.g. The proof of Fermat's last theorem.
My sense of philosophy - after reading maybe a dozen of the classics, was there was little that was accepted as true. Yes, there are self-consistent chains of reasoning but the foundation blocks are more a matter of "taste", and the amount of rigor in the chains varied - complicated by the fact that human language is inherently not very precise.
True, but that is because philosophy doesn't deal with answers. It deals with a more fundamental question, the question of truth itself. Instead of providing a framework for finding singular answers to exact formulations like mathematics does, philosophy ultimately provides a framework for exploring and expanding the limits of certainty.
Please remember that the entire STEM field grew out of the questions of those philosophers you dismiss as "no clear answers": our entire scientific process (empirical theory-building) is based on the previous explorations of philosophers on truth and knowing. Similarly, both capitalism and Marxism grew out of the questions of philosophers about the structure of society.
You need questions before you can answer them. Philosophy deals with the questions. Our answers to those questions have become the main pillars of modern society.
Mathematics is also based on foundations which are in some sense a matter of taste e.g. Euclidean vs non-Euclidean geometries. IMO at a certain level of abstraction mathematics and philosophy are the same thing.
This seems like a poor curriculum rather than an actual flaw in philosophy, without the logic framework philosophy is just the history of ideas. If taught in a manner similar to math, where you learn logic tools and then build on top of them over time, I would expect similar and overlapping development.
A philosophy class teaching fundamentals in logic - taught in a way similar to math - is probably more like math in the sense of the article than philosophy in the sense of the grandparent post. The grandparent probably meant something more like a class on ethics, which doesn't really have the same style of beauty and truth that math does. This is certainly my impression after taking a few philosophy classes and a ton of math ones, including a philosophical logic class cross-listed with math.
This isn't a flaw of philosophy. Lots of fields of human study are great. Math just has some things that make it unique. But just like defining what art is, it's hard to pin down exactly what it all is.
Philosophy, as it is typically taught, contains too much historical annotation and attribution to my taste. I want the factual/logical content only, not the "Wittgenstein said this" and the "Nietzsche said that", because it almost implies that I have to choose a side.
I doubt it is because there might be no such core.
(Or rather, if you believe in the core, you are automatically a analytic philosopher? A school I have many sympathies for, but hardly the only or even mainstream of philosophy.)
For me, philosophy shows the ideals of human affairs and the the higher ideals, but it seems to suffer from too many opinions and contradictory viewpoints.
Mathematics gives you the kernel of the universe. Its a much finer grained tool that represents absolutes.
That the heat from your laptop flows in this way thoughout the air around it. As does the warmth of breath, or body heat or the heat surrounding an open flame?
And then there are logical principles that create "elegant" logical structures like in group theory, where you develop two different ways of looking at things only to find out they were one and the same all along (e.g. Lagrange's theorem).
I agree that Philosophy is important, and probably more so as you say, but a love of Philosophy and an appreciation for Mathematics is hard to beat.
Philosophy can invite discord due to the circular nature of using language to define language with the intention that this is supposed to bring clarity to our reasoning. Not to mention that the name "Philosophy" is often co-opted by charlatans to advance their brands.
But Philosophy coupled with Mathematics brings a kind of calm that's removed from language and an appreciation for simple, provable, truths.
I would like to add that in most cases it is crucial to understand philosophy behind everything. Whatever you try to learn, if you don't understand the way of thinking, don't see a roadmap, then you don't understand a subject.
In schools and colleges this is not always the case that we learn philosophy of a subject. Especially in schools. This is sad.
Why is it so hard for people to accept that it's probably a good thing not everybody wants to do exactly the same things as they do? Do you really think that if everybody focused their life on any single intellectual venture you picked, we'd live in a better (or even functioning) world? Or that people who enjoy X and don't like Y (or simply don't like it as much!) should just deal with being unhappy so you can conclude that Y is the One True Intellectual Pursuit?
I don't think that's the argument being made. But that by learning X or Y, you can add a lot of substance or perspectives to your life that you may not have had before. These 'things' more likely than not can bring many positives to aspects of your life that aren't related.
Both come from the latin "Gratia" (favor), but you can thank the extremely ambiguous prefix "in" for the confusion. Depending on the word it can act both as "into" as well as "without".
> Instead, I'd been forced to waste years on things like math, biology, etc.
Is this sarcasm?
If not, can you outline "the supposed benefits of doing math" versus the benefits of studying philosophy, that are "far more applicable to most people's everyday lives"?
I've heard a similar argument before but I'm skeptical. As a science-trained person with only a little exposure to philosophy, while philosophy uses rational arguments, etc., it seems like it's all too easy to bullshit your way out of.
My question: what does graduating with a degree in philosophy say about one's abilities, and how can you show it? (I would argue the types of problems that an upper-level science/math/CS person is expected to solve very clearly shows analytical and creative ability to anyone with a little knowledge of the subject)
The way I see it, most subjects in school focus on teaching you what to think. Philosophy focuses more on how to think.
That's obviously not true in all cases (there are plenty of dogmatic philosophies) but in my opinion it's much more common for people to teach things like math, biology, history, etc. as a series of objective, memorized facts and formulas. Doing that with philosophy is far more difficult because so much of it is subjective and relies on criticism and analysis. In that regard, learning philosophy serves as a foundation for learning all other subjects, or learning anything in life, for that matter.
I also personally believe there's more beauty in great philosophy than in any poem or song. There are plenty of Socratic dialogues, stoic passages, and political pamphlets that still give me chills when I read them.
Based on your comment I'm going to assume you've never taken a mathematics course above Calculus. Anything beyond that is the absolute antithesis of "memorized facts and formulas".
It also fails to apply to any sciences I'm aware of at the research level. Philosophy pretty much only exists at the research level, but it's also a much smaller pool of jobs than most fields.
> Doing that with philosophy is far more difficult because so much of it is subjective and relies on criticism and analysis.
Any education in a topic worth it's salt should include that once you get passed the intro courses (because the intro courses are often necessary to provide context for your analysis). Whether curricula succeed in doing so, or students engage themselves enough to do so, is another matter.
What is the philosophy behind arrow functions in JavaScript? Should I just use them blindly, or should I understand why this exists and why/when I should use it? What problem does it solve? Why?!
In my opinion every article about some aspect of programming should begin what an explanation of the philosophy behind it's development. Why does this exist? What problem does it solve? When should it be used/not used?
Totally disagree. Philosophie try to find answer to question which can't be answered. It's just a lot of energy wasted in unsolvable problems. And it is far easier. In philosophy there is no truth, everyone can be right.
Math is a lot tougher, and so, teach you discipline and rigor much better.
Su seems almost blind to the fact that you can do math outside an academic environment. You don't need to go to school for math to experience truth, beauty, play, etc in the context of math. You can read books, play around with the results, read articles on places like Quanta. You probably won't discover new things this way, but it will enrich your life.
If your goal is for math to make people's lives better, then assuming school is involved at all is another unnecessary restriction.
Su opened his talk with the story of Christopher, an inmate serving a long sentence for armed robbery who had begun to teach himself math from textbooks he had ordered. After seven years in prison, during which he studied algebra, trigonometry, geometry and calculus, he wrote to Su asking for advice on how to continue his work. After Su told this story, he asked the packed ballroom at the Marriott Marquis, his voice breaking: “When you think of who does mathematics, do you think of Christopher?”
It doesn't sound like he has that bias to me. He opens his talk speaking about someone who is a non-traditional student doing match outside an academic environment.
In an ideal world I might have continued in academia but the career path is so twisted you have to either be insanely good (at math and at managing time) or just hate yourself enough to sacrifice your best years working essentially in the metaphorical darkness well outside the spotlight and most likely alone and poorly paid.