Philosophical movements in the 17th century tried to mimic the geometrical method of the ancients. Some saw Euclid—with his ruler and compass in hand—as a “doer,” and thus characterised geometry as a “maker’s knowledge.” Others got into a feud about what to do when Euclid was at odds with Aristotle. Descartes thought Euclid’s axioms should be justified via theology.

Transcript

Everybody has seen the painting The School of Athens, the famous Renaissance fresco by Rafael. It shows all the great thinkers of antiquity engaged in lively intellectual activity. Plato and Aristotle are debating the relative merits of the world of ideas and the world of the senses, both gesticulating to emphasize their point. Others are absorbed in other debates and lectures, somebody’s reading, somebody’s writing.

But here’s something most people don’t notice in this painting. There is one and only one person in this entire pantheon who is actually making something. Everybody is thinking, arguing, reading, writing. Except Euclid. Euclid is drawing with his compass. He is producing the subject matter he is studying. He is active with his hands. He’s practically a craftsman among all these philosophers.

In the ancient world, the mathematician is the maker. Geometry is the most hands-on of all the branches of philosophy and higher learning.

Today the cliche is that a math nerd is almost comically feeble in anything having to do with physical action.

But ancient geometry was in the thick of the action. You had to roll up your sleeves to do geometry. Even in theoretical geometry you would constantly draw, construct, work with instruments. It was a short step to engineering. The greatest ancient mathematician, Archimedes, is almost as famous for his feats in engineering. Such as mechanical devices for lifting and moving heavy objects, and for transporting water. Archimedes and other mathematicians were also at the front lines of war, building catapults and many other warfare machines according to precise calculations. They were architects. The Hagia Sophia in Istanbul for example, was designed by a mathematician, Isidore, who had written an appendix to Euclid’s Elements.

In early modern modern times, like the 17th century, this link between mathematics and concrete action was well understood and appreciated.

Francis Bacon was sick of traditional philosophy because “it can talk, but it cannot generate.” This frustration led him to the radical counterproposal: to know is to do. “What in operation is most useful, that in knowledge is most true.” And on the other hand “to study or feign inactive principles of things is the part of those who would sow talk and nourish disputations.” So we have to condemn much traditional philosophy and turn more to action, to doing.

Perhaps the most important difference between ancient mathematics and ancient philosophy is precisely this. That mathematics is active, while philosophy merely “sows talk and nourishes disputations.” Perhaps that is the explanation for why mathematics proved so fruitful, still thousands of years later, both for intricate theory, such as planetary motions, and for practice, such as engineering, navigation, and so on. Try doing that with Aristotle’s doctrine of causes or Plato’s theory of the soul. Those things are great for “sowing disputations” but if doing is the goal then you can’t get much mileage out of them.

Thomas Hobbes, another famous 17th-century philosopher, very much agreed with this analysis. Hobbes famously declared that “Geometry is the only science that it hath pleased God hitherto to bestow on mankind.” How so? What makes geometry different from all other branches of philosophy and science?

Constructions, of course. Hobbes is very explicit about this. “If the first principles contain not the generation of the subject, there can be nothing demonstrated as it ought to be.” This is what makes mathematics different. Its principles contain the generation of the subject: Euclid’s postulates correspond to ruler and compass, and these are tools that generate the figures that geometry is about.

All philosophical and scientific theories are based on some assumptions or axioms. But they are not generative axioms. They are not a recipe for producing everything the theory talks about from nothing.

In this light we can readily appreciate for instance Hobbes’s otherwise peculiar-sounding claim that political philosophy, rather than physics or astronomy, is the field of knowledge most susceptible to mathematical rigour. Here’s how he puts it:

“Of arts, some are demonstrable, others indemonstrable; and demonstrable are those the construction of the subject whereof is in the power of the artist himself, who, in his demonstration, does no more but deduce the consequences of his own operation. The reason whereof is this, that the science of every subject is derived from a precognition of the causes, generation, and construction of the same; and consequently where the causes are known, there is place for demonstration, but not where the causes are to seek for. Geometry therefore is demonstrable, for the lines and figures from which we reason are drawn and described by ourselves; and civil philosophy is demonstrable, because we make the commonwealth ourselves.”

As bizarre as this may sound to modern ears, it makes perfect sense when we keep in mind the all-important role of constructions in classical geometry.

Indeed there are many things that only the person who made it truly understand. At this time, the 17th century, various mechanical devices were becoming more common. Such as pocket watches and all kinds of other machines based on gears and cogwheels and so on. The person who made it knows what all the parts are for, but an outsider cannot see this very easily at all. Today another example might be computer programs. The person who wrote it knows how it works, what it can do, how it could be changed, what might cause it to fail, and so on. It would be very difficult for someone else to get a similar sense of how it all works, even if they had access to the code, or they could pop the hood and look at the gears so to speak. Only the maker truly knows: “maker’s knowledge” is a slogan often repeated in the 17th century.

Hobbes took this idea and built a general philosophy from it. His general philosophical program can be read as a direct generalisation of the constructivist precept to the domain of general philosophy. Here’s how Hobbes defines philosophy: “Philosophy is such knowledge of effects or appearances as we acquire by true [reasoning] from their causes or generation.” This is basically a direct equivalent in more general terms of the principle that constructions are the source of mathematical knowledge and meaning.

Indeed, Hobbes explicitly draws out this parallel: “How the knowledge of any effect may be gotten from the knowledge of the generation thereof, may easily be understood by the example of a circle: for if there be set before us a plain figure, having, as near as may be, the figure of a circle, we cannot possibly perceive by sense whether it be a true circle or no. [But if] it be known that the figure was made by the circumduction of a body whereof one end remained unmoved” then the properties of a circle become evident. You understand a circle because you make it, in other words.

Another way of putting it is that “The subject of Philosophy, or the matter it treats of, is every body of which we can conceive any generation.” Just as, classically, the domain of geometry is the set of all constructible figures.

Concepts that are not constructively defined can easily be contradictory or meaningless: a common problem outside of geometry. As Hobbes says: “senseless and insignificant language cannot be avoided by those that will teach philosophy without having first attained great knowledge in geometry.”

Again, as we have discussed before, anchoring geometrical entities in physical reality is a warrant of consistency. Hobbes makes this point as well. “Nature itself cannot err”; that is to say, physical experiences “are not subject to absurdity.”

It is notable that Hobbes and other 17th-century thinkers who invoked geometry did not have in mind simple school geometry and some superficial remarks in Plato or Aristotle. Rather, they were referring to the rich picture of the geometrical method that emerges from a thorough study of advanced Greek geometry and technical writers. When they call upon geometry they mean not some simplistic idea of axiomatic-deductive method but a rich methodology only conveyed implicitly in the finest ancient works of advanced geometry.

This is why the constructive aspect shines through so clearly. It’s importance is evident if you study the mathematicians and build your idea of philosophy of mathematics from there. You’re not going to learn anything about that by reading Plato and Aristotle.

Hobbes is very clear about this. As he says, his philosophy of geometry is “to an attentive reader versed in the demonstrations of mathematicians without any offensive novelty.” Indeed, one must be “an attentive reader,” because one must draw out the philosophical implications left implicit in these sources. And one must be “versed in the demonstrations of mathematicians,” meaning the technical Greek authors. As Hobbes calls them, those “very skillful masters in the most distant ages: above all in geometry Euclid, Archimedes, Apollonius, Pappus, and others from ancient Greece.” This is why Hobbes, in one of his works, “thought it fit to admonish the reader that he take into his hands the works of Euclid, Archimedes, Apollonius, and others.”

Many other 17th-century philosophers picked up the same themes. Some took it to the epistemological extreme of