I’ve just come back from a few weeks in Thailand, where I gave talks at Chulalongkorn University, Thammasat University and Chiang Mai University. I also attended a conference on Geometry and Graphics in Bangkok, and met my mathematician friend Paolo Bertozzini, who is always a pleasure to talk to, full of insights and anecdotes from his long experience in the Land of Smiles. By the way, Thailand is a fascinating and wonderful place, visit if you can!
Travelling gives me time to muse; I can’t always be at the computer (even though on this particular trip I did spend a lot of time working on two papers with my graduate students) and waiting for airplanes or sitting in them gives one time to speculate. A topic that I have been pondering is: to what extent are mathematicians scientists? Or are we actually something else?
Probably I am steering towards the something else. Sure, mathematics and science have a lot in common. Science uses lots of maths first of all in setting up its theories. This used to be much more true of the physical sciences, but increasingly the biological sciences are also becoming more mathematical–or at least some aspects of them are. And some applied mathematicians are pretty close to being physicists, but not really experimental ones. I believe a strong case could be made that most mathematicians do research like scientists: we observe patterns, try to formulate theories to explain them, and then subject those theories to experiments–in our case calculations–to discover that they are probably wrong and need to be modified.
But in my experience this somewhat standard view-point misses an important distinction that needs a historical perspective to appreciate. Mathematics has been around a lot longer than science. The Greeks were doing mathematics at a very high level more than two thousand years ago. Mathematicians have a long sense of history, humbled and awed by the great minds which have preceeded us, of accomplishments in centuries gone by which we can no longer hope to surpass, or even equal.
The scientist thinks, and feels, quite differently. Science really only kicked off about 500 years ago in Europe, when people slowly started thinking thoughts like: how do we really know when something is true? Can belief and truth be separated? Does our desire that the world be a certain way prevent us from seeing it as it really is?
I remind you of the answers to these kinds of questions that people came slowly to appreciate: that the source of all true knowledge is observation: careful, unbiased and thorough. From the observations we make, we formulate theories to explain them. The simplest and most powerful theories take precedence. Finally we examine the implications and predictions of our theories and see if these are born out. If so, we strengthen our faith in our theories but do not become dogmatic about them. We are prepared to be wrong, and to change our minds when confronted with new evidence and explanations.
How much deep knowledge and power resides in the understanding which I have crudely summarized in the previous paragraph! A whole brave new way of thinking, of seeing, of understanding the world. Brave, because we are prepared to face the music, however it may sound. No longer must the heavens dance to a tune of our liking. Maybe we are small, and insignificant, and weak. But we will have the courage to admit it, and to carry on none-the-less in understanding the world, unconcerned if we are no longer at the center of God’s great plan, should such a One exist. And we are not upset if our theories overturn and disprove the thoughts of a previous generation–in fact we welcome such, and strive towards the breakthrough that upturns the applecart.
I believe that modern mathematics has lost its way logically, and that a new and far more interesting mathematics awaits us. I have a fair amount of evidence to support this point of view. Rational trigonometry gives a much simpler and more powerful approach to trigonometry and geometry, making computations easier—but the kicker is that it actually makes logical sense, as opposed to classical trigonometry, the development of which is a logical basketcase! And Universal hyperbolic geometry is likewise a complete logical overhaul of hyperbolic geometry, again replacing pictures and wishful thinkings with simpler and much more careful reasonings. Both of these new developments result in many novel and beautiful theorems.
It has been interesting, and I will admit somewhat (but not overly) disappointing, to see how uninterested my fellow pure mathematicians are in contemplating really new directions of thinking, and how unsure they are in applying their own critical analysis to weigh the evidence, rather than rely on authority and precedence.
The force of habit in people’s thinking weighs heavily on them, the mark of a heavy and bloated subject. How can I inject more scientific thinking amongst my fellow pure mathematicians? How can I make the subject lighter? These are the kinds of thoughts I have been thinking in Thailand.
I have wandered on either side of whatever line exists between mathematicians and scientists, though I am still a graduate student and therefore cannot be considered authoritative. I can give you my thoughts on the similarities and differences between the subject, and my overall impression of the sociology of modern mathematics and why I chose not to be a part of it. This little essay isn’t going to be especially coherent, since I am still formulating many of these thoughts and am always ready to change my mind in the light of new reasoning.
Mathematicians and scientists in modern times share a very important trait: they are professionals. That is, anyone who identifies herself as a mathematician or a scientist is saying something about their source of income. A professional needs to get paid for what they do, so they necessarily have a conservative attitude to their professional activities. Neither a functional analyst nor a molecular biologist will want to hear that the entire foundation of their subject might be flawed, because they might be out of a job! Even if they’re not out of a job, they might find out that their entire career has been based on lies — not a pleasant situation for anyone.
I think the hallmark of scientific thinking is falsifiability. In science, the goal is to make testable predictions. In mathematics, by contrast, falsehood is a rather nebulous concept. All of the reasoning I encountered during my time learning mathematics was essentially Aristotelian: as long as you accept the truth of certain (supposedly basic) premises and the rules of the game of logical inference, there is no ability to contradict the results of this game. As you have taught me, Dr. Wildberger, there are good reasons to doubt the supposed foundations of the subject (like the axiom of infinity).
I left mathematics because I saw it as the province of people whose heads were stuck in the clouds. Leaving aside the question of the value of the subject, I saw a lot of people who wanted to avoid a great part of human society and stick to highly abstract pursuits that they proudly claimed had no intrinsic usefulness! This was made very clear to me when I read Hardy’s “A Mathematician’s Apology”, which I consider to be one of the most ill-conceived essays in the history of humanity. It is all well and good to pursue a subject purely for the love of the subject, but it is not reasonable to expect to get paid with tax dollars without some kind of social justification for the work.
I don’t think this about every mathematician, and I don’t think that mathematics is unique in this way. I do think, however, that it makes the activities of most mathematicians quite opposed to the goal of most scientists. A scientist is explicitly trying to advance humanity’s understanding of nature; that is, natural phenomena that can potentially be exploited to improve the lot of humanity. We live in an age where the benefits of this process are undeniable.
What is the motivation for mathematics? As far as I can tell from having met some mathematicians, it is to solve interesting puzzles. I think that’s a bad justification for the field of human thought that I love best. My own perception is that mathematics is not so different from theoretical physics: mathematics is the study of what is possible with certain instruments. Watching your lectures on geometry has caused me to rethink the whole foundation of the subject. Your emphasis on the fact that projective geometry comes from the use of a straight-edge, affine geometry from the ruler, Euclidean geometry from the straight-edge and compass, and so on make me think of geometry as fundamentally a theory of tools. Because I study quantum physics now, this kind of thinking resonates quite strongly with me. If mathematicians were to accept that this is indeed the purpose of their work, they might find themselves thinking far more scientifically than they do now.
Thanks for the interesting thoughts.
That was a great comment to read.
What you are describing is what I call cultural momentum. I have the same critique when one tries to introduce a new paradigm of programming in the IT community. When a community is already streaming in a particular direction the force of habit is hard to break. You can not easily steer them into another direction. It is hard to stop the momentum.
I also think Kernel is correct in observation. I sat to lunch with an acquaintance after we attended a logic seminar. This fellow’s specialty is Philosophy of Mathematics. In passing he said, there is no way he can be hired to do his work. No one in academia wants to hire hime. Perhaps he was like me who did undergrad work in 70’s or 80’s when idealism permeated that generation. I believe that idealism affected mathematicians too. Kernel is right in leaving mathematics. If I had known it would be this difficult to get a job in mathematics, I would have followed my father’s advice, I would have gone to stats or engineering. Idealism does not pay.
Idealism does not pay because practical people are in majority so they decide about tax dolars. Even more, they do not like to pay taxes. Materialism consumes peoples minds and that makes real problem solvers rare. Every small business is calculating their prices according to price of gasoline, but nobody give tax money to the guys who invented wheel. They are forgotten long ago. Minkowski did not create his theory with intention to be used in Einstein’s one. It just got used, not sold. It is hard to make money with knowledge. Because it is considered to belong to everybody. Knowledge and education are the sole driving force of huge advancement in technology that we witness. It is accelerating with every year. Now practical people want to influence education so that it generate even more revenue. But they never had a clue how it works.
I think that you got the results you did in part because you had a different reason for getting involved in math then most mathematicians. I think most professionals earned a degree for the purpose of cashing in on it. It might seem cynical, but thats just my impression, because it seems most professionals don’t care about their profession. If I really love what I’m doing, I simply get a different result then if Im just doing something for the money. Further, if Im just doing something just because its fun, I get a different result then if I have a grand vision. Its not that doing things for fun, or making money are bad. They are essential! But they should be balanced with these other more long term sustaining inclinations.
Sadly, maybe most other mathemeticians don’t care about your findings because they just don’t care that much about mathematics to begin with. Maybe making it through the rounds of learning math once was a tramatic experience, and they just don’t want to go through it agian. Maybe, after getting a degree, cashing in on it, and then realizing that the degree in and of itself doesn’t make you any smarter makes a person feel really insecure. Maybe they don’t like the idea of making math more accessible. People take on peculiar behaviors when they are under alot of stress. Maybe they feel they are protecting their territory.
I came just by chance, while browsing on web, to your interesting tangential thoughts.
I agree with many of your ideas, and disagree with a lot of other ones as well.
For instance, I also believe that “modern mathematics has lost its way logically, and that a new and far more interesting mathematics awaits us”.
I am quite sure that mathematics is the only one “real science”, according to the original meaning of the word science = scientia in Latin.
Definition of SCIENTIA
: knowledge, science; esp : knowledge based on demonstrable and reproducible data .
In my language – Slovak, science = VEDA, which is just a different form of the werb VEDIET with the meaning to know. For me science = knowledge. But what really is knowledge?
Is it something stable? Is it not changing with the overall development of humankind? Is it perhabs a well defined phenomenon or rather a process converging to some universal truth ? Or is this process diverging to infinity, a generic chaos, where everything is possible under certain circumstances ?
What I find as the most magnetic feature of mathematics and doing mathematics is that unbiased desire to reveal the truth, to understand how “things” work, a really urgent need, almost killing urge to get the mesage and uncover the hidden beauty of the “rules governing the chaos of life”.
Are you coming to Croatia this September?
Nice to hear from you, and thanks for your thoughts. As for Croatia in Sept, I probably, sadly, cannot make it.
Next year I hope to!
I kinda like this comment of Daniela, it enhances my view of mathematics. At any rate, to be a good scientist, one must be aware of the limitations of her/his craft.
I find it interesting how you say that modern mathematics has lost its way logically, Ive watched quite a few of your videos on you tube (thank you) and I love the way you challenge the order of the day as it were. You also talk about how observation played a key role in the development of scientific thought. Unfortunately the peer review system is now being used to make sure that observations and the conclusions from them remain unpublished if they go against ‘accepted truth’. There is also a massive problem with observation, as a mathematician once convinced me, you shouldn’t believe anything that you see, feel, hear etc, because we don’t really even know what existence is.
In a similar direction, we might say that what we see, feel, hear etc is such a small part of the totality of all existence, and such an intrinsically skewed part, that we should be very careful about drawing wide ranging conclusions from it. Why should the world ultimately be understandable to us? Why should even a significant portion of it be understandable to us? I think the complexity and richness of mathematics is a suggestion that we are biologically far from being able to handle the `whole truth’, whatever that might be. So modesty, and a willingness to entertain myriad possibilities, ought to accompany our forays into scientific and mathematical discoveries.
In my opinion, modern sciences, like modern math, are also “contaminated” by some “strange” philosohical thoughts, which regards science as merely a theoretic system that can explain something we observe (as much as possible), and the systems are not unique, there is no rights or wrongs in choosing whichever system we like.
This may due to the “death of philosophy” claimed by Wittgenstein, where (possibly, I’m not so sure, since I’m not a philosopher) he deconstructed the concept “absolute correcteness”. So whatever plausible are correct, if it can have reason.
I don’t know whether this is a good philosophical thought or it is not.
I think you are spot on.
There are a lot of obvious, and a gazillion of less obvious but relatively easily avoidable intellectual mistakes that current scientific/intellectual elites do not even attempt to avoid. To name a few: Goodhart’s law, the invalidity of predeterminism or block universes, that “perception is not action”, confusing the map with the territory. Not to mention their constant bigotry towards anyone who doesn’t just swallow whatever they remember from high school.
The most annoying thing is not even the death of philosophy, but the death of science, since nowadays everyone seems to assume that science is literally just collecting data and fitting functions to it. You can see it happen again with the new telescope in several months. I bet they will claim again they found a lot of new things, and they will just introduce even more complicated (and often incorrectly derived) algorithms to massage the data into whatever their 200 year old models says they should see.
Thanks Ashnur for the nice comment. A related difficulty is that in mathematics even a basic scientific orientation is often missing: appeals to authority seem to count for a lot more in our discipline — especially in pure maths.
Why should the world ultimately be understandable to us? Why should even a significant portion of it be understandable to us?
I like these questions. Mathematicians should engage in this type of questions. It seems pretty relevant and interesting to ask these philosophical questions.
I conjecture that we want to have an understandable universe because humanly speaking, we want to have some control of it. We can not control what we do not understand.
Can I ask another question on the side? I am wondering the motivating forces why people go into maths? In my case it was a form of escape. There was a psychological factor perhaps 50% of it. Now that I am much older, I do not fear anymore the truth that it was best for me to be a nerd as I was not socially skilful as a teenager. It provided a vehicle to while away my time not having much friends.
Mathematicians are not scientists, because they work with a created idealized world. For their work, it seems not so important if this world exists beyond of us (in Plato’s kingdom of ideas) or if it exists in our heads only (e.g. Brower more recently).
Mathematical ideas come often from observations of the real world, of course, but mathematics has freed itself from reality already since the times of Plato’s geometers. If mathematicians should work in a math world with infinity or in a math world without infinity, as you want it, can be debated, but what would be the criteria for a choice? Correspondence to the real world should not be an important criterium, I think, but usefulness of math models for the real world should be.
Who says “work with a created idealized world” ?? Have you ever heard a professional mathematician make that statement publically? Do you believe that the great mathematicians prior to the 20th century would buy that?
“created idealized world” are my words in my comment. An example is Plato’s world of ideas that contained perfect lines and circles, which was an idealized world. For its discussion it seems secondary to me if that world of ideas exists independent of humans, was created by God or in the Big Bang, or is created by human minds in every lifetime anew. Primary is that its elements are not limited to phenomena existing in the physical world. Like a good story is not limited to events having happened in the physical world.