To Live Your Best Life, Do Mathematics
quantamagazine.org
quantamagazine.org
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.
this is true throughout the sciences.
it's a big issue.
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.
The former is a thrill to many, and the latter is at best a chore for all.
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.
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 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 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.
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.
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.
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.
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.
The Wikipedia article has a long section of criticism of Elsevier: https://en.wikipedia.org/wiki/Elsevier#Criticism_and_controv...
But that said, I don't know of any author that wouldn't give you a gratis copy, oh, ok, maybe an old 'draft' that is 'close' to the publication.
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?
I think many more papers would be read if authors invested more time in learning how to write.
This seems, at first, to be a problem in academia, but, at further glance, is really a problem with education.
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.
>Teaching mathematics shouldn’t be about sending everybody to a Ph.D. program. That’s a very narrow view of what it means to do mathematics.
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.
(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)
... 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.
(From TFA.)
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.
It took mathematics much beyond what was known at the time to get anywhere close to a solution.
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"
[0] https://mathtourist.blogspot.com.au/2010/06/tiling-with-pent...
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.
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.
However... being a math major is really different from a career in academia. People really should be encouraged to major in math.
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)
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.
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).
Later on, things get better, yes, but not much.
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.
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.
^ THIS.^
If math education could only be more like: https://betterexplained.com
Feedback needn't be a necessity in a work. Assessments and criticism will come if the work is essential, perhaps many years in the future.
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.
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."
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.
[1] https://www.wolframalpha.com/input/?i=sum+of+1%2F(4%5Ek)
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.
"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.
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.
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.
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.
All the more reason to open the floodgates so we can get the few people with both skillsets.
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.
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.
Nothing wrong about it, just an observation.
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.
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.
But starting with both, I never met one.
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.
They bear part of the responsibility.
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.
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.
As usual, generality is not reality.
But this is my experience of it.
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.
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.
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.
I do agree that math can and should be widely appreciated.
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.
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 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.)
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 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.
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.
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.
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.
https://mathyawp.wordpress.com/2017/01/08/mathematics-for-hu...
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.
> “Why program by hand in five days what you can spend five years of your life automating?”
As to your response... Should inmates be deprived of all distractions, especially trivial ones? How often do you BuzzFeed or candy crush or Facebook?
I'm definitely an outlier here, sample size one. :-)
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.).
Unless you count Madame Bovary.
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.).
<play, beauty, truth, justice and love.
Math and music gives me all those.
My Math inclination brings tears To binary situations and lasts (as in the last time I'll ....)
Truth is a set.
Justice is relativity (<,>,=)
Play is math humor with the pun as king.
And love is random uncontrolable feeling.
http://www.ccs.neu.edu/home/matthias/HtDP2e/part_preface.htm...
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.
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?
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.
Chapter VI in this short book of dialogues, for example: http://www.gutenberg.org/files/17490/17490-h/17490-h.htm
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.
In schools and colleges this is not always the case that we learn philosophy of a subject. Especially in schools. This is sad.
Of course, a one size approach won't fit everybody.
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.
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.
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 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 guess we'll just have to completely disagree on that.
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?
And then to spell out the proof? Oof
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.
http://rationalwiki.org/wiki/Salem_Hypothesis
http://scienceblogs.com/tfk/2007/11/11/the-salem-hypothesis-...
http://www.theness.com/index.php/creationists-mechanical-eng...
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/
Harder numbers:
http://www.nuff.ox.ac.uk/users/gambetta/Engineers%20of%20Jih...
(expanded book form) http://press.princeton.edu/titles/10656.html
(less numbers, but foundational)
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.
[1] https://groups.google.com/forum/#!topic/talk.origins/Xunl5Sl...
Does common philosophical discourse generally attempt this though?
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 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
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/
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.
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.
A lot of 'continental' philosophy seems to be a mix of historical analysis, and cult-of-personality/lit-crit...
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.
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.
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.
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).
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.
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.
Mathematics gives you the kernel of the universe. Its a much finer grained tool that represents absolutes.
Can you look at the heat equation and see the beauty in it? https://en.wikipedia.org/wiki/Heat_equation
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).
(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.)
Math is a lot tougher, and so, teach you discipline and rigor much better.
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)
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 didn't realize chinese was a person of color.
If your goal is for math to make people's lives better, then assuming school is involved at all is another unnecessary restriction.
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.