Opinionated History of Mathematics

Rationalism says mathematical knowledge comes from within, from pure thought; empiricism that it comes from without, from experience and observation. Rationalism led Kepler to look for divine design in the universe, and Descartes to reduce all mechanical phenomena to contact mechanics and all curves in geometry to instrumental generation. Empiricism led Newton to ignore the cause of gravity and dismiss the foundational importance of constructions in geometry.

Transcript

Here’s a fundamental problem in the philosophy of mathematics. You can sit in an isolated room, in an arm chair, and prove theorems about triangles, such as the angle sum of a triangle or the Pythagorean theorem. When you do this, you have the feeling that you have established these results with absolute certainty. You feel that they must be true because of how compelling the proof is. And you feel that you have established this by thought alone, by purely intellectual means.

Mathematics is unique in this respect. In other subjects, thinking is a powerful tool, but it is always supplemented by observation and experience. If you spent your whole life isolated in a locked room, you would not be able to say anything about the laws of astronomy or the anatomy of the digestive system, because without observation, with only pure thought, it is impossible to even get started in those field. But you could figure out everything about triangles. If one day you were released from your prison where you had been sitting for decades, you could go out and measure actual triangles and you would find that, indeed, their angle sum is always two right angles, the Pythagorean theorem always holds for right-angle triangles and so on. Just as you had predicted by pure thought.

This is a bit of a mystery. Because it shows that there are two sides of mathematics that are difficult to reconcile. On the one hand, the internal, mental conviction that mathematics establishes absolute truths purely by reasoning. On the other hand, the external, physical fact that mathematics works in the real world.

What is the bridge between these two worlds? It is as if there is a natural harmony between our minds and the outer world. What is the cause of that harmony?

These two poles can be called rationalism and empiricism. Rationalism takes mathematics to be fundamentally a matter of pure thought. This fits well with the sense we have when doing mathematics, when reading Euclid, that we are establishing absolute truths by sheer reasoning. But it doesn’t explain why mathematics works so well in the physical world.

We have encountered some rationalists already: Plato, Descartes. We saw how Descartes solved the problem. Mathematics is pure thought, and it works in the physical world because the Creator put mathematical ideas in our minds. As the Bible says, “God created man in his image.” That is to say, God created the world based on mathematical ideas, and then created humans and sort of pre-programmed their minds with the same kinds of ideas that he had used to create the world.

So no wonder there’s a harmony between the mental and the physical worlds: they both stem from the same source, the Creator, who used the same principles when designing both. Descartes said basically this quite explicitly, as we recall. Plato pretty much hints at the same idea. God is a mathematician. That is a central belief in Platonist thought as well. And it is a necessary thesis for the rationalists to explain why mathematics works so well.

We have already encountered some empiricist as well: Aristotle, Francis Bacon. They think knowledge ultimately comes from the world around us. From that point of view, it is no mystery that mathematics works on physical triangles. It stems from physical experience to begin with, so of course it conforms to physical experience.

The challenge for the empiricists is instead to explain the mental experience of doing mathematics; our feeling that it brings absolute truth by pure thought in a way that no other subject does. From the empiricist point of view, this feeling is a mistake, a delusion. We think we are doing pure thought, but actually mathematical thought is generalized experience. We think we can sit in a closed room, an arm chair, and figure things out about an outside world that we have never even seen. But it only feels that way.

We have seen and touched many lines and triangles and squares our entire life, since the year we were born. We have internalized this experience. It has become second nature to us. Basic truths of geometry, such as Euclid’s axioms, may feel like core intuitions that are much more pure and absolute and undoubtable than things we know from experience. But that feeling is a delusion, according to the empiricists. Our minds, our feelings have imperfect self-awareness. Just as we are not aware through introspection how our digestive system works, so we are not conscious of the psychological origins of our mathematical intuitions.

I think we can agree that rationalism and empiricism both face big challenges. The challenge for rationalism is to explain why mathematics applies to the physical world. Traditional rationalism had an answer that was very compelling at the time: the explanation in terms of God, the Creator. But nowadays we may want an atheistic answer. And then rationalism is back to square one, facing the original problem all over again, without any solution in sight.

Empiricism doesn’t have that problem, but it has other ones. If mathematics comes from experience, how can it seem so absolute and undoubtable? How can an exact science come from inexact sensory impressions? If mathematics is based on experience like everything else, why does it seem to be such a different kind of knowledge in so many respects? Those are challenges for the empiricist to answer.

It matters how you answer these questions. It shapes the kind of science that you do.

Consider for instance Kepler, the 17th-century astronomer. He was another rationalist. As Kepler says: “Nature loves [mathematical] relationships in everything. They are also loved by the intellect of man who is an image of the Creator.” That’s almost word for word how I described the rationalist position just moments ago.

Kepler felt that the world was designed with the intent that we should study the universe mathematically. As he says: “Whenever I consider in my thoughts the beautiful order [of the universe] then it is as though I had read a divine text, written onto the world itself saying: Man, stretch thy reason hither, so that thou mayest comprehend these things.”

In fact, scientific facts support this view, in Kepler’s opinion. For example, as he says, “Sun and moon have the same apparent sizes, so that the eclipses, one of the spectacles arranged by the Creator for instructing observing creatures in the orbital relations of the sun and the moon, can occur.”

That is indeed a striking fact: that the moon is exactly the right size to precisely block out the sun at the moment of a solar eclipse. From the point of view of modern science, this is a remarkable coincidence. It’s pure chance that the moon is exactly the right size.

You can understand why the explanation in terms of purpose was more compelling in Kepler’s time. Witnessing a solar eclipse is a spiritual experience. It all seems so perfect. Much too perfect to chalk it up to chance. It’s very disappointing that modern science offers nothing more than this non-explanation of such an emotionally compelling spectacle.

And not just modern science. Such views were around already in Kepler’s time. Atomism is a classical worldview that is indeed happy to attribute almost everything, eclipses included, to chance and randomness. According to Kepler’s teacher, Melanchthon, such views “wage war against human nature, which was clearly founded to understand divine things.”

So here we have again that double challenge to empiricism. If mathematics is just one type of knowledge among many that we pick up from experience, then, first of all, why does the universe show so many signs of being mathematically designed? Like the thing with the eclipses, but there are also countless other examples one could use to make this point. Empiricism has no answer to this. It thinks that’s all just a bunch of coincidences, and we are just fooling ourselves by looking for purpose and design that isn’t there.

And secondly, if empiricism is right, and mathematics is just experiential knowledge like everything else, then why does mathematical reasoning feel so uniquely compelling and convincing? As Melanchthon says, mathematics is as natural to a human being as “swimming to a fish or singing to a nightingale.” Just as animals are born with these instincts, so our minds are innately predisposed to do mathematics. Empiricism does not explain why that is the case, or why that seems to be the case.

So it’s understandable that Kepler was a convinced rationalist instead. And this conviction shaped his scientific work. Astronomers are “priests of the book of nature,” as Kepler said. So he was always looking for meaning and purpose and design.

For example, the telescope was a new invention in Kepler’s time, and it was a big moment when the moons of Jupiter were discovered. Kepler immediately looked for the purpose behind the existence of these moons. He concluded that Jupiter must be inhabited. Why else would it have moons? As Kepler says: “For whose sake, the question arises, if there are no people on Jupiter to behold this wonderfully varied display with their own eyes? We deduce with the highest degree of probability that Jupiter is inhabited.”

Another of Kepler’s attempts at uncovering divine design was his theory of planetary distances. According to Kepler, the Creator had chosen the number and position of the planets according to a very beautiful and pleasing mathematical design. Namely, a plan based on the five regular polyhedra.

Euclid discusses the regular polyhedra at length in the Elements. There are precisely five of them, as Euclid indeed proves in the very last theorem of the Elements.

Kepler figured God was as fascinated by these shapes as Euclid had been. So when God asked himself how many planets there should be in the solar system, and how far from the sun to put them, God figured that the most mathematically pleasing way would be to choose six planets, and to have the spaces between them chosen in such a way that the five regular polyhedra fit between them like a nesting doll.

Kepler’s theory in fact fit the data very well. You could calculate planetary distances from astronomical measurements, and you could calculate size proportions of the regular polyhedra from Euclid’s Elements. If you put these things side by side in two columns they come out remarkably close to one another.

So again Kepler explained things that modern science doesn’t explain at all. Why are there six planets? Why are they positioned at those particular distances form the sun? Why does the moon fit precisely on top of the sun during an eclipse?

Kepler explained all of these things. If you accept the basic outlook that it makes sense to think of the creator of the universe as a Geometer, then Kepler’s explanations are very good. This is Kepler, the best mathematical astronomer of his age. These are not some whimsical religious musings. It’s very serious science. Very good science, one might argue.

Meanwhile, modern science doesn’t explain any of these things. There is no explanation, there is no why, according to modern science, of course. It’s all just chance. The solar system was formed by a bunch of random rocks getting caught in a gravitational field. Whatever positions they took up is just random.

It’s easy for us to judge Kepler. But shouldn’t science explain more things as it develops? Not fewer things. You would think that science should take things that are not explained and explain them. Instead of taking things that are already explained and attributing them to coincidence instead. And yet that is precisely what happened when Kepler’s theories were abandoned.

In any case, this Kepler stuff is interesting for all kinds of reasons, but for our purposes, what I wanted to show was that it matters whether you are a rationalist or an empiricist. Rationalism, as we saw, almost requires the hypothesis that God was a Geometer, just as Plato and Descartes and Kepler all said. And that assumption has major implications for how you practice mathematical science. It suggests looking for deliberate design put into the world by a mind that is essentially like our mind, as far as mathematics is concerned.

So that’s one way in which the rationalism-empiricism divide strongly shaped scientific practice in the early 17th century. But that was not the end of it. Here’s another example: the contrasting ways in which Descartes and Newton approached cubic curves.

Cubic curves are the next step beyond conic sections. Conic sections are curves of degree 2. They were studied in great depth by the Greeks. Cubic curves are called cubic because that have degree 3. So they are the more complicated cousins of the conic sections. In the 17th century, this was natural direction to take geometry: to understand curves of degree 3 and higher in the same depth that the Greeks had understood conic sections.

For instance, conic sections come in three classes: ellipse, parabola, hyperbola. Can one find an analogous way of classifying cubic curves? There are going to be more classes because cubics are more complicated. But maybe with the right principle of taxonomy one can impose order among their variety in way that is as useful as the division into ellipse, parabola, and hyperbola is in the theory of conics.

Newton did precisely this. He gave a very detailed and advanced technical study in which he classified cubic curves in several different ways. He divided cubic curves into “species” as he says. That’s Newton’s own term, and it’s a vivid one.

Taxonomising curves into “species” makes Newton sound like a pioneering explorer-scientist forging into unknown jungles and studying all the strange creatures. When you find a new exotic insect, you put it under a microscope and study all its properties. How many legs does it have, how many eggs does it lay, and so on. It’s the same when studying curves. How many crossing points, how many inflections points, and so on. It’s the zoology of mathematics.

This metaphor fits very well with the epistemological ideals of empiricism. You learn by studying the great diversity of things out there. Into the jungle! That’s the call of empiricism. That’s how you learn things. By immersing yourself in the unknown.

“The best geologist is one who has seen the most rocks.” That’s another slogan of empiricism. Experience is the source of knowledge, in other words. If you want to understand rocks, you need to look at a whole lot of rocks. And if you want to understand cubic curves, you need to look at a whole lot of cubic curves, first of all. Once you have built up a store of experience, then maybe you will see some patterns starting to emerge and you can begin the process of systematising or taxonomising the “rocks.”

Empiricism is all about diving in at the deep end and figuring it out as you go. This corresponds to reading Euclid backwards. You start with the complicated stuff, the Pythagorean theorem and such things. Those kinds of things are the exotic beasts that you encounter “in the jungle.” Gradually, you seek to bring order into the chaos by finding general principles that account for the phenomena you observe.

That’s empiricism. And it’s completely backwards according to rationalism. That’s not how you learn things. You can’t start with observations, with the phenomena. Perception is unreliable. Aimless exploration unguided by the intellect is bound to be a waste of time leading nowhere.

The way to knowledge is thinking. To “meditate,” as people used to say. You have heard of Descartes’s Meditations. That’s even the title of one of his works. The source of knowledge is meditation. That is to say, deep thought where you basically close yourself off from the world. Sitting in an armchair in a closed room. That’s where you make progress in understanding, not running around in the jungle.

So Newton’s way to study cubic curves was the empiricist way. Get your machete out and start chopping your way through the thick of it. Eventually you become familiar with all these wild things you encounter, and you start to see what kinds of species there are and how they are related.

Descartes was the opposite of this. A rationalist. Descartes studied cubic curves too, but through meditation. His big book is La Géométrie (1637). He doesn’t study cubics specifically, but all algebraic curves. So curves of any degree, not just degree 3.

Already we see a typical rationalist characteristic: rationalism starts from the general; empiricism starts from the specific.

Rationalists withdraw into meditation because they do not trust individual observations. Thought is more reliable. If you sit back in an armchair and introspect about what is knowable, you are bound to come up with very general and abstract truths: I think therefore I am; the whole is greater than the part; two lines cannot enclose a space. Gradually, you have to work your way from there, step by step, to any specific fact you need to explain. Just as Euclid gradually works his way up to more and more complex and detailed material by starting with very general principles that ultimately entail all the rest.

So the rationalist in interested in all-encompassing abstract law or axioms. It is important to the rationalist that all truths can in principle be deduced from these axioms. But it’s less important to actually do this. The rationalist is most interested in the fundamental axioms or laws, because those are the source of the certainty of knowledge. The specifics derived from them merely inherit their certainty from the certainty of these foundational axioms.

So the very first principles of the entire field is where you need to focus your attention if you are a great rationalist philosopher. And that’s exactly what Descartes does in his book, La Géométrie. Even the title fits with this point of view: The Geometry; it’s a very total, definitive account of geometry as a whole, just as the rationalist epistemological ideal demands.

This is further confirmed in the very first sentence of the text: “All the problems of geometry …”––that’s how Descartes opens his book. He starts with extreme generality, just as rationalism suggests one should. He wants to find the principles that can be used to solve “all the problems of geometry,” in principle.

Descartes doesn’t care so much about the details. He is very keen to explain why his principles are sufficient to solve “all the problems of geometry,” but has very little patience for actually solving any of those problems. This is reflected in the very last sentence of his book.

Descartes writes: “I hope that posterity will judge me kindly, not only as to what I have explained, but also as to what I have intentionally omitted so as to leave to others the pleasure of discovery.”

This is a bit dishonest, of course. He did not omit the details merely out of kindness to the reader, obviously. His focus on the general and lack of interest in the specific is a consequence of his rationalist outlook.

Newton is the opposite. He loves the details; he loves getting stuck in with some obscure technical problem. In fact, his long treatise on cubic curves is full of technical details but he gives very little attention to explaining any general conclusions. It’s hard to see the forest for the trees.

That’s good empiricism, of course. Rationalism thinks you can trust specific results because they are derived from reliable general principles. The certainty of knowledge resides in the axioms, the general principles. That’s where you need to focus your attention to secure the rigour and reliability of reasoning. And that’s what Descartes does.

Empiricism looks at it the other way around. It is the details, the little things, that are the most knowable. Knowledge starts from the directly observed phenomena, with all their specificity. That’s the root of reliability and certainty. Abstract principles are trustworthy only insofar as they are inferred from a large body of facts.

It’s the same in physics. To Newton, the empiricist, the starting points are specific facts. The orbital time of Jupiter, the speed of Saturn. Specific observable facts. You have to start there and then infer general laws like the law of gravity by showing that it fits a long list of facts. It is the specific facts that give credibility to the general law.

Not so to Descartes. The introspective, meditative, rationalist way of doing physics is to figure out first what properties of moving bodies are the most undoubtable. What are the things that are like Euclid’s axioms, but for mechanics?

Descartes did physics exactly this way. In his view, the most undoubtable core principles of physics are the laws of collision of two bodies. If one body bumps into another, what happens? Well, if one is twice as heavy but they have the same speed, then so-and-so happens; if one is twice as heavy but the other is twice as fast, then so-and-so happens; etc. Those are the kinds of principles that Descartes thought one could establish through pure thought and meditation.

Descartes saw this as analogous to Euclid’s geometry. Euclid’s axioms are about lines and circles: the basic building blocks of all geometrical figures. More complex figures are built up from there by combinations of lines and circle, or ruler and compass. In the same way, in physics, complex phenomena can be regarded as ultimately generated by the simple root phenomenon of the collision of two bodies.

Indeed, modern science kind of agrees about that part. If you exhale on a cold day, you breath forms a cloud that moves in complex ways. It seems to flow or float, but really it’s just lots and lots of tiny molecules crashing into each other millions of times, and that gives rise to this kind of flowing pattern that you see on a larger scale.

So simple generative principles can be enough to account for all kinds of things behind their immediate reach, though elaborate repeated composition. Just as lines and circles kind of “give birth” to all geometry, including very complicated shapes that aren’t just round or straight.

Actually, lines and circles are not enough to generate all geometry. They can’t generate cubic curves for example. Descartes is very interested in this issue. And indeed, in his book La Géométrie, he supplements the ruler and compass with another basic generative principle for drawing curves. A kind of linkage principle. You can build a sort of machine that consist of multiple rulers and pegs interlinked in certain ways, and as you push one part of the machine the other parts move in specific ways because of how all the parts are interconnected. An ordinary compass is sort two rulers nailed together. In the same way you can make more elaborate devices composed of more rulers. This gives rise to “new compasses,” as Descartes calls them. And these are sufficient to encompass “all the problems of geometry,” according to Descartes.

In a way it might seem contradictory that it was the rationalists, like Descartes and Leibniz, who were so concerned with the making of geometrical figures with concrete devices. Shouldn’t a proper rationalist hate physical instruments, like Plato did?

But there is no contradiction. Descartes cared about geometrical instruments for theoretical reasons. As I just emphasised, constructions in geometry go naturally with the general rationalist idea of the mind generating all knowledge from within itself. It’s a form of self-reliance. It doesn’t need anything from the outside world.

And earlier we have spoken about how constructions are connected to the epistemological foundations of geometry. Maker’s knowledge. Constructions are the most knowable thing, and the most secure form of geometrical knowledge, protected against many threats of paradoxes and contradictions. So that’s another way in which constructions go well with rationalism, which is of course very much concerned with what are the most undoubtably knowable things.

So these instruments like the ruler and compass and the generalisations of them that Descartes conceived are theoretical, not practical. There’s a funny anecdote that sums this up in the Brief Lives by Aubrey—a late 17th-century collection of biographical stories, maybe not super reliable exactly but this story could very well be true. Here’s what this biographer Aubrey says:

“[Descartes] was so learned that all learned men made visits to him, and many of them would desire him to show them his instruments. He would drawe out a little drawer under his table, and show them a paire of Compasses with one of the legges broken: and then, for his ruler, he used a sheet of paper folded double.”

Quite amusing, and it fits with what I said about the constructions being theoretical.

So we see that the idea of drawing curves with instruments in geometry is analogous to the idea of explaining all of physics in terms of collisions of little bodies. They are both simple, intuitable principles that generate the entire world of phenomena.

From a rationalist point of view, you need such principles. You start in the simple and pure world of meditation and you need to reason your way to the complicated and messy outside world. So you need a bridge that goes from the simple to the complex. Contact mechanics is such a bridge in physics, and ruler and compass is such a bridge in geometry.

But this is only necessary if you are a rationalist. If you insist on starting with pure intuition and thought, then you need such a bridge to the phenomena and the outside world.

But if you are an empiricist you take the outside world—the jungle—for granted as given, as a starting point, so you don’t need to explain how it can be generated by repeated composition of simple principles.

Indeed, Newton rejects both contact mechanics and geometrical constructions at the same time, for precisely this reason.

The fact that these two things are intimately related is not lost on Newton. This is why he starts his big masterpiece on physics by talking about the construction of line and circle in geometry. A very weird way to start a physics treatise to modern eyes, but it makes perfect sense if we keep in mind the background of Descartes and rationalism and everything I just outlined.

I’m referring to Newton’s Principia of 1687. Descartes was long dead by then, but his ideas about the foundations of physics were as relevant as ever. Leibniz, who was a contemporary of Newton, was a rationalist like Descartes. Like Descartes, Leibniz attached great importance to contact mechanics in physics and constructions in geometry.

So when Newton’s Principia came out, Leibniz was very upset that Newton had abandoned the principle of contact mechanics, which was so essential to the entire rationalist worldview. Let me quote Leibniz on this point. Here’s what he said: “A body is never moved naturally except by another body that touches and pushes it. Any other kind of operation on bodies in either miraculous or imaginary.”

Newtonian gravity is precisely one such “other operation”; something that cannot be explained in terms of particles bumping into one another. This is why Leibniz condemns very fiercely the notion of gravity as a foundational principle of physics: “I maintain that the attraction of bodies is a miraculous thing, since it cannot be explained by the nature of bodies.”

That is to say, Newton’s law of gravity cannot be explained or arrived at from a rationalist point of view. Newton in fact agreed. If anything, he makes this point in even stronger terms than Leibniz. Here’s what he says: “It is inconceivable that inanimate brute matter should, without the mediation of something else, which is not material, operate upon and affect other matter without mutual contact, as it must be, if gravitation be essential and inherent in it. That gravity should be innate, inherent, and essential to matter, so that one body may act upon another at a distance through a vacuum without the mediation of anything else, is to me so great an absurdity, that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it.”

Very strong words there from Newton. And we can understand why. He wants to discard the rationalist outlook entirely. He is not interested in winning broad support for his theory by trying to argue that it sort of fits with rationalism somehow. He could have given that a shot. He clashed with many influential people: Descartes, Huygens, Leibniz. He could have tried to go a diplomatic route and try to come up with reasons for why his way of doing science was compatible with their rationalist commitments. But he chose not to. This is why he comes on so strongly in these quotes about how gravity is rationally inconceivable and so on.

In this way, Newton moves the conflict into the area of rationalism versus empiricism generally, instead of arguing about the interpretation or meaning of gravity specifically. “With the cause of gravity I meddle not,” says Newton, since “I have so little fancy to things of this nature.”

So what Newton wants to justify is not gravity specifically, but a the empiricist way of doing science generally, in which you don’t care about such questions at all. Questions such as how to give a rationalistic account of gravity, or explaining how a meditating mind in an armchair could arrive at the necessity of the law of universal gravitation. Those questions should simply be ignored, says Newton. Which makes sense from an empiricist points of view, but is sheer madness from a rationalist point of view.

So Newton bites the bullet on the cause of gravity. He says: yeah, I know my physics completely clashes with the core beliefs and methodology of rationalism, but rationalism is wrong anyway.

Now, as I said, the role of contact mechanics in physics is analogous to the role of constructions in geometry. Newton knows this, and this is why, to justify his physics, he starts by talking about how to interpret the role of constructions in geometry. Here is what he says right at the beginning of the Principia:

“The description of right lines and circles, upon which Geometry is founded, belongs to Mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn. For it requires that the learner should first be taught to describe these accurately, before he enters upon Geometry; then it shews how by these operations problems may be solved. To describe right lines and circles are problems, but not geometrical problems. The solution of these problems is required from Mechanics; and by geometry the use of them, when so solved, is shewn. And it is the glory of Geometry that from those few principles, fetched from without, it is able to produce so many things. Therefore Geometry is founded in mechanical practice.”

So that’s a clearly empiricist account of geometry. Not only because it obviously grounds geometry in the physical world, in physical practice and experience. But also because it takes away the idea that the axioms need to be justified by being intuitive and undoubtable. That was important to the rationalists, but Newton does away with that.

This is how Newton can justify that he “meddle not with the cause of gravity.” Geometry likewise doesn’t “meddle” with the construction of curves, but merely postulates their description—in fact, geometry postulates these things precisely “because it knows not how to teach the mode of effection,” just as physics does not know how to teach the cause of gravity.

So Newton has twisted Euclid into support for his physics. This is why the preface to the Principia is about constructions in geometry, such as the ruler and compass of Euclid. If geometry doesn’t really know how to generate these curves, but only takes them for granted and goes from there, then physics can do the same with gravity.

So Newton and Leibniz clashed along such lines. And not only them. One could argue that there’s a geographical element to this divide. Empiricism is to some extent a British movement more generally: not only Newton but also Francis Bacon, John Locke, Wallis—just to name some people we have already encountered before. Meanwhile, Leibniz’s rationalistic tendencies in his science and mathematics were shared by his leading colleagues in Continental Europe, such as Descartes and Huygens.

By way of summary, let me read a passage by Newton on his scientific method, and I will insert comments on how what he says fits exactly with what we have discussed. The passage begins:

“As in mathematics, so in natural philosophy, …”

Already very interesting. In other words, Newton is announcing that his scientific method is based on the method of mathematics; the method of Euclid basically. Ok, so what is this this methodological principle that is common to both mathematics and science? The sentence continues:

“… the investigation of difficult things by the method of analysis, ought ever to precede the method of composition.”

Analysis corresponds to reading Euclid backwards. To analyse is to break down into smaller pieces. Composition corresponds to reading Euclid forwards. To compose is to put simpler pieces together to form more complex results. Newton continues:

“Analysis consists in making experiments and observations, and in drawing general conclusions from them by induction. By this way of analysis we may proceed from compounds to ingredients, and from motions to the forces producing them; and in general, from effects to their causes, and from particular causes to more general ones, till the argument end in the most general.”

“General” indeed: a key words here. From observations, that is to say from specific facts, one infers more general underlying principles. Empiricism goes from the specific to the general; rationalism the other way around.

It is also nice that Newton mentions that analysis goes “from compounds to ingredients”: this is precisely the chemistry or cooking metaphor that we used before when discussing how to read Euclid backwards.

Newton continues:

“This is the method of analysis, and the synthesis consists in assuming the causes discovered, and established as principles, and by them explaining the phenomena proceeding from them, and proving the explanations.”

That is to say, reading Euclid forwards is of course also essential. The method of analysis that the empiricist uses does not dispense with this directions of Euclid; it merely reveals that a preliminary stage is necessary to understand its meaning. It is through the preliminary analysis, the backwards direction of reasoning, that one arrives at the principles—not by direct intuition, as the rationalists would have it. Then the forward direction, the composition or synthesis, proves that these principles really work; that is to say, that they are sufficient to prove everything. That part is the same to both rationalists and empiricists. The key difference is how they account for where the principles or axioms came from.

So those are Newton’s own words, corresponding very closely to the story I have told. Of course Newton and Leibniz and all these guys were acutely aware of all of this. In this way they were much more philosophically conscious than most scientists of later ages. And clearly it shaped their science very profoundly, as I have shown by several examples. So that’s all the more reason to keep pursuing these questions. As indeed we will.