The comet we know as Halley’s Comet had been spotted many times before the great English astronomer Edmond Halley (1656—1742), but it was not recognized as a periodic comet until eighteenth century Europe, which is significant. The Chinese had apparently never calculated the orbits of either Halley’s Comet or other comets which they had observed. They had a large mass of observational data, yet never used it to deduct mathematical theories about the movement of planets and comets similar to what Kepler, Newton and others did in Europe. Newton’s Principia was written a few generations after the introduction of the telescope, which makes it seductively simple to believe that the theory of universal gravity was somehow the logical conclusion of telescopic astronomy. Yet this is not at all the case.What would have happened if the telescope had been invented in China? Would we then have had a Chinese Newton? This is impossible to say for certain, of course, but I doubt it. Chinese culture never placed much emphasis on law, either in human form, as in secular Roman law, natural law or divine law. If the Chinese had invented the telescope, I suspect they would have used it to study comets, craters on the Moon etc., which would clearly have been valuable, no doubt. Any culture that used telescopes would have generated new knowledge with the device, but not necessarily a law of universal gravity.- - - - - - - - -
From the fourteenth until the twentieth century, almost all important global advances in mathematics were European. I would be tempted to say that European leadership was stronger in mathematics than in almost any other scholarly discipline. Perhaps the simplest explanation for why the Scientific Revolution happened in Europe is because the book of nature is written in the language of mathematics, as Galileo once famously stated, and Europeans did more than any other civilization to develop or discover the vocabulary of this language.
The introduction of the telescope was a major watershed in the history of astronomy, but we should remember that it alone did not create modern astronomy. The birth of astrophysics in the late nineteenth century came through the combination of the telescope with photography and spectroscopy, all inventions that were exclusively made in Europe. Spectroscopy could not be developed until chemistry as a scientific discipline had been formed, which only happened in Europe. New fuels, engines and materials later made space travel possible. Asian rockets were powered by gunpowder and weighed a couple of kilograms at most. They could not have challenged the Earth’s gravity and explored the Solar System. The Saturn V rocket that launched Apollo 11 on its journey to the Moon in 1969 used liquid hydrogen and oxygen, elements which had been discovered in Europe. The very concept of gravity, too, was developed only in Europe. The exploration of the Solar System and the universe at large was to an overwhelming degree made possible by a single civilization alone, the Western one.
Read the rest at the Brussels Journal.
16 comments:
Fjordman
Mathematics is a language whose vocabulary one invents rather then discovers. Mathematics is thus an invention of the human mind and resides therein. It is the equivalent of abstract art - there is nothing like it in the universe. It is coincidental that it is useful for describing some aspects of nature.
Another aspect of the scientific discovery process that Europe took on, is that it happened only when Christianity became the sole faith or guiding philosophy of Europe. Neither the Greeks or the Romans took to this road though they had atleast a 1000 year headstart in mathematics.
DP111: Organized science was the creation of the Christian European civilization of the early modern age, although it had roots back into the medieval age, for instance with the establishment of the first universities. Capitalism was also the creation of late medieval and early modern Europe. It did not exist in Greco-Roman Antiquity. Europeans did make active use of Greek mathematics and natural philosophy as well as Roman law, but the synthesis they made was unique and was not anticipated by the Greeks or anybody else. The Romans did exactly zero useful work in mathematics. The Chinese did at least create some good mathematicians.
DP111 wrote "It is coincidental that (mathematics) is useful for describing some aspects of nature."
- hmm, well those coincidences does seem to occur rather often, don´t they?
Seriously, when I read a sentence like that, I always wonder "why?", what is the motivation behind uttering an absolute falsehood? Does man experience some kind of pleasure by stating the excact opposite of the truth? A pleasure he can mistake for the pleasure we take in understanding judgements that are true and evident?
Giving this some thought lately I have come to the conclusion that this might actually be the case. Consider statements like "the isrealies are the true nazis" or "Islam is the religion of peace", "in reality mathematics is a kind of abstract painting"; "the national socialists where really ultra liberals". "the rapist is actually unable to cope with his sexual anxiety" etc. etc.
Hearing such sentences, one experiences a kind of mild surprise, which does resemble emotions obtained by evident judgements. This leads to what might be termed "emotional contra-evidentiality disorder"...
...Anyway with regards to chinese versus european mathematics, as I also stated in comments to your latest essay, I believe the most fundamental difference lies in the Pythagorean cult, and the heritage they left. Because although the chinese had something similar to pythagoras theorem quite early on, they valued it completely different. For them it was a nice discovery, and a great way to measure land etc. just like mulitplication can be seen as a fast way of counting.
But this is all very far from the pythagoreans, who claimed that numbers were the only thing that really exists, and that all other things are governed by their inherent relations. This meant that to the pythagoreans math was not a technique it was a matter od life and death. It is this almost religious obsession numbers and selfevident judgements, which gave the desire to deliver a demonstrable proof of your insights and treasure aciomatic statements such as "the shortest distance between two points is a straight line" as the highest good, as did the mathematikhoi over centuries.
DP111:
"Mathematics is a language whose vocabulary one invents rather then discovers. Mathematics is thus an invention of the human mind and resides therein. It is the equivalent of abstract art - there is nothing like it in the universe. It is coincidental that it is useful for describing some aspects of nature."
Mathematics does not consist solely of its vocabulary. The logic, the rules, and the analysis of mathematics are much more important than its vocabulary. Thus, mathematics is not an invention of the human mind and it is more than just a coincidence that it is useful for discovering aspects of nature.
The mathematics of calculus and differential equation enabled the Apollo 11 astronauts to travel from eath to the moon and back. The same mathematics also enabled the Apollo 12 astronauts to travel from earth to the moon and back. It is not a coincidence that it worked both times.
I very much value the greatness of Newton, Leibniz, Einstein, Euler, DesCartes, and so many countless great European scientific minds.
Truly, the world would not be what it is today without their marvelous achievements.
We Indians have also made assorted contributions to math and science. Among them:
- the Base 10 numeral system invented in Southern India
- algebra was likewise not invented by the Arabs, but again in India
- Damascus Steel was not invented in Damascus, but in India as Wootz Steel
- Satyendra Bose extrapolated quantum bosonic behaviour to gas molecules
- molecular photonic scattering, known as the Raman Effect, after Nobel Laureate CV Raman
- Chess also comes from India, and we still like to play it today
But I don't feel that any of us can afford to sit on our laurels and repose on past feats of accomplishment by previous generations. We all have a responsibility to each make ourselves into the next 'greatest generation'.
Love of science is certainly something which distinguishes intellectually appreciative cultures from the retrograde ones, and should therefore be something which unites forward-looking peoples.
Jungle Jim
You are right that not just the vacabalary but the rules of logic are just as important in mathematics. Both of them go hand in hand. However, logic itself is a construct of the human mind. Now if we argue that the construct of the human mind, and also thought itself, is a product of a "universal logic", then yes, mathematics itself is part of a greater truth of the universe that we discover with the human mind.
There is though no reason to assume such a thing. Logic, and mathematics by extension, is a product of the human mind - and hence an invention.
It is worth noting that Riemann's work on non-Euclidean surfaces was considered pure mathematics. It was only later that Einstein used some of that work in general relativity. However, Riemann's work did not arise from any desire to explain any phenomenon in the physical world - it was pure abstract thinking, or playfullness of human curiosity. The same can be said for complex analysis, and even calculas. It is so with much of the body of mathematics. It may be that in the future, some of the vast body of mathematics will find scientific application - but that is coincidental, while others may never be.
Fjordman
You state "Organized science was the creation of the Christian European civilization of the early modern age, although it had roots back into the medieval age, for instance with the establishment of the first universities".
Agreed.
Now, here's the great irony of the age. DP111's characterisation of logic as nothing more than a product of the mind is one of the tenets of reductionism - that the mind is nothing more than the sum of its squishy grey parts, and that logic by extension is ultimately nothing more than the product of blind chance.
Yet, many scientists working in the very forefront of advanced physics consider that there is an underlying logical structure to the entire universe. According to them we use abstract mathematics to describe this structure but our minds didn't create it - it is a product of the universe itself. A universal logic or rationality.
The irony is that htese people are closer to the christian standpoint than might appear obvious. A christian, educated and devout, will tell you that that our rationality stems from a higher rationality, that our limited logic extends from an unlimited logic.
To someone like me the idea of emergent rationality is absurd. By its very nature it has no foundation and so, to reach the logical conclusion of the argument - that our rationality is a product of the human mind and nothing more than the end-point of a series of random chances - is to conclude that our rationality is irrational. But if you conclude that our rationality is irrational, what basis do you have for reaching that conclusion? In teps subjectivism and up stands post-modernism.'
Our mathematics may be the product of our minds but they are describing something that isn't. Kepler's laws, Newton's laws and Einstein's laws are descriptions of actual phenomena. The rules - the physics - so described by each exist independent of the language used to describe them.
Graham
You misunderstand me if you think that I ascribe that the mind is simply the sum of its squishy parts. And from that you carry on that I deduce that “logic by extension is ultimately nothing more than the product of blind chance”. Never did I infer this or ascribe blind chance to the order and structure of the human mind and thought.
And yes I know that many scientists “working in the very forefront of advanced physics consider that there is an underlying logical structure to the entire universe”. Indeed they do. It is though a matter of faith - and it is good faith, for it leads to continued and hopeful search for that underlying truth. But that is not what I said. I made a statement about mathematics and not physics. Further, I also noted that “Now if we argue that the construct of the human mind, and also thought itself, is a product of a "universal logic", then yes, mathematics itself is part of a greater truth of the universe that we discover with the human mind”. In effect, the human mind reflects a universal truth. But this is conjecture. If it turns out to be so, it opens the possibility that if we ever meet up with aliens, their minds too will reflect that universal truth, and hence the possibility of some reasonable understanding between two species who may be quite different biologically. Despite Hollywood scenarios, I have doubts.
You wrote, “The irony is that these people are closer to the Christian standpoint than might appear obvious. A Christian, educated and devout, will tell you that that our rationality stems from a higher rationality, that our limited logic extends from an unlimited logic”.
Excellent. Newton, Dalton, James Clerk Maxwell, Mendel, that is, all the very greatest scientists of the world, those who totally changed the way we view this world, will agree. Oh and Fred Hoyle too. Thank you.
Look forward to your thoughts.
DP111 and Graham --
As a mathematician by education, training, and sometime employment, I feel compelled to weigh in.
Mathematics as a discipline is a descriptive syntax bound by well-defined and clearly formulated rules. Those rules may apply to the observed world, and may owe their formulation to observation of real events, but they are independent of the observed world. Any mathematical corpus begins with a set of postulates — statements taken as given — and derives the rest of the system from them through the application of consistent rules.
However, mathematics does depend on something that lies outside of itself: human language. The symbols, rules, and formulations must be described and transmitted through speech and writing. In the end, mathematics owes its structure and origins to the speech centers (Broca’s Area) in the parietal lobe of the brain. It depends on our instinctive ability to generate distinct concepts, and to formulate rules to govern their interaction; in a word, the formation of a grammar.
In that sense, mathematics is a subset of grammar, or at least the “deep structure” of speech. It’s accurate to say that no aspect of mathematics has ever been created, transmitted, or applied without the mediating role of language.
P.S.
A corollary to the above is that there may be something about the particular manifestation of “deep structure” as encoded in the Indo-European languages that aided the formation of mathematics as we know it today.
Notice that besides its well-known European sources, mathematics owes much of its advanced formulations to Persia and Hindu India, both of which developed their accomplishments using Indo-European languages.
Contrary to popular mythology, the Arabic-speaking world contributed very little actual innovation to the field.
If it turns out to be so, it opens the possibility that if we ever meet up with aliens, their minds too will reflect that universal truth, and hence the possibility of some reasonable understanding between two species who may be quite different biologically. Despite Hollywood scenarios, I have doubts.
Given the inability for people on this same planet to understand the motivations of others, when our minds are ostensibly made of the same Ousia, the same "stuff", the mere fact of mind stemming from a universal underlying rationality doesn't necessarily lead to the inevitability of those minds having even a remote chance of communication with each other.
I think our miscommunication might stem from the possibility that we're discussing different things. You're talking about mathematics the language, whereas I'm thinking more of the underlying nature of the universe described by that language. Mathematics imparts universal truths but it's descriptive of those truths, not the truth itself. Our own language is inadequate for properly describing it, in the same way that it cannot really impart the meaning of blue. You can describe blue things but you just try and explain what blueness is and see how far you get.
Which is probably the point you were trying to make, come to think of it...
Perhaps part of the problem from my point of view came in the fact that the language you were using is strikingly similar to the reductionist arguments of mind. If you were playing devils advocate in that, then I'm afraid you were far too subtle for me. If your intention was other than reductionism then again, too subtle for me... I'm not, by nature, a very subtle mind. :)
(And you forgot Hooke.)
To me, it is crucial to distinguish between mathematics and philosophy of mathematics. The former is a rigorous science, based on selfevident axioms and derived procedures in a coherent system. This is the exact opposite of philosophy of mathematics, which is not a science in any way, but is more of a dispute, where people claim one reason or another for these axioms being/seeming so true. The philosophers give elaborate criticisms of each others work, but have all failed to deliver proof of their respective hypothesis.
To me this is all good, if it wasn´t for the fact that this failure of philosophers often leads people to claim, that since the self evident cannot be proven, it must be a mere postulate. Hence all math is but a mere assumption. Nothing could be further from the truth. These axioms are every bit as true, as the millions of conclusions which are based upon them. The only reason they hasn´t been proven, is because philosophers have demanded a special kind of proof or "evidence" than is to be expected from the self evident (for instance the attempt to "deduce" it from something else, although this something always seemed less evident).
Secondly I am very sceptical of the shift made by Wittgenstein from the logical to the “grammatical”, and believe it was made out of the frustrating search for some kind of "special" truth. Atleast it seems weird that wittgenstein would expect people to believe his assumptions, since he delivers no proof for them in any manner. In fact it seems odd that a man, that doesn´t even accept the fact that 2 times 2 is 4, even tries to convince anybody.
I sometimes believe that wittgensteins move from logic to grammar, was the second blow that killed western philosophy. The first one was made by Kant, when he decided to give up on the world and confine himself to theory of consciousness and foundational theory of science. Because as I said above, theory of science is not scientific, and theory of consciousness, is not rigorous either, and is not conscious of the laws of nature, which should be the main interest for a person loving knowledge.
The first blow made the student unsure of himself, the second made him unsure of knowledge itself. Atleast to me it cannot be a coincidence, that the most decadent post-modern philosophies of our time all exchange reality with conscioussness or discourse and logic with linguistics?
I am sure wittgenstein did not intend this, he probably just wanted to “stay hungry” and not take any answer for granted, but alas, the consequenses have been devastating. Leaving the heritage of the proud english enlightenment, lame in their defeat of the french disease in the eighties and nineties. Giving way to such disturbing problems as for instance political correctness, which is the prime example of linguistics taking the place of logic...
On a sidenote, this discussion somehow reminded me that I wanted to add Dan Dennetts "breaking the spell" to the videoteque. It´s a great lecture sketching out his attempt to devise a scientific research of religions as "systems of thought" underlying some sort of survival of the fittest scheme, very interesting, particularly when applied to the religion of the desert. Highly recommended.
Dan Dennett
BB posted: Mathematics as a discipline is a descriptive syntax bound by well defined and clearly formulated rules. Those rules may apply to the observed world, and may owe their formulation to observation of real events, but they are independent of the observed world.
And that essentially was the basis of my gentle hint to Fjordman that mathematics is not "discovered" but "invented". It well be that Fjordman wrote "discovered" carelessly or in haste. The amount of Fjordman's written work is so huge, it is forgivable. Also in one of my previous posts I wrote -It( referring to mathematics)is coincidental that it is useful for describing some aspects of nature. I'm glad that you see this.
Your description of mathematics though, makes it look cold, cut and dried. There is something else that is generally overlooked -intuition. And this is not subject to mathematical logic or the dictat of syntax. I regard mathematics as abstract art of the very highest order - subject on the one hand to logic and syntax, and on the other to nothing but the general playfulness, disordered and chaotic at times, of the human mind. It is from this playground that new art arrives.
Please forgive me if I have hijacked the thread away from the main thrust of your blog.
Graham posted: I think our miscommunication might stem from the possibility that we're discussing different things. You're talking about mathematics the language, whereas I'm thinking more of the underlying nature of the universe described by that language.
Yes. I was sticking to mathematics, and trying to steer away from physics and all that. But once the subject is in, it is hard not to put one's two cents. Though still glad that it went that way.
Graham posted; Given the inability for people on this same planet to understand the motivations of others, when our minds are ostensibly made of the same Ousia, the same "stuff"
Great you saw that, for I was waiting with this in my armoury.
And Graham, I have to be honest and say that I was playing devils advocate, though I had hoped that I wouldnt be caught out. There is a reason for this, in this most confused of times in which we live.
Graham, Jungle Jim, and all
You have written much of value. I shall be away for a day on "duties" that cannot be avoided. Will look in if I can.
DP111 --
Your description of mathematics though, makes it look cold, cut and dried. There is something else that is generally overlooked -intuition.
Actually, that’s just my discursive style of writing, to which I tend to resort when matters get technical. I’m actually quite passionate about matters mathematical, as my wife and son can attest!
Intuition is rarely overlooked by any mathematician who is good at what he does. We all know how important it is. The solution usually emerges in a flash of intuitive insight, which is inevitably followed by hours of drudgery creating a rigorous proof of what is so obviously true.
The same is true when debugging computer programs. I have a knack for that particular unpleasant activity, and I have learned to trust my intuition, because it is right in at least 90% of the cases.
DP11 again, I have to object to the underlying equivocations of your post. To claim that the objects of mathematics are independent of the observed world, does not entail that the application of mathematics is a mere coincidence. The independency of the mathematicians field of study has been aserted by countless philosophers, from pythagoras to frege, who unanimously reject any idea of coincindentiality. On the contrary, the separation from empirical observation of the real world is meant to secure the absolute necessity of mathematics, which eo ipso secures its applicabillity in the real world. I don´t remember reading any philosopher who claimed the existence of ideal objects as an argument for coincidentiality.As far as I know this idea followed in the footsteps of russels paradoxes, the fall of set-theory and wittgensteins skepticism. Much like the quantum mechanics of niels bohr, is often used to claim the "subjectivity" of physics.
This necessity is also the reason that most people find it more natural to describe advances in mathematics, as "discoveries" unlike technological "inventions". If you find out that the shortest distance between to points is a straight line, you don´t feel like you are being creative, making stuff up, because you know, deep in your heart that this was true all along. too claim otherwise, would be like saying that newton "invented" the spherical planet earth.
Lastly with regard to the cheerfulness of math, it may not be quite as common as one could hope. Many of the greatest mathematicians of our age ended up commiting suicide in insanity asylums. Both Cantor, Boltzmann, Gödel and Turing all pushed themselves to the edge, and tipped over. BBC made an interesting documentary about these mathematicians some yaers ago. It is called "dangerous knowledge".
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