History of Science & Technology Q&A (June 3, 2026)

**Stephen Wolfram** (0:01)
Hello everyone, welcome to another Q&A about History of Science and Technology. And I see a whole bunch of questions saved up here.
Let's see, there's one from Robot. How far back does the idea of discrete space go, and who first took it seriously as a scientific possibility?
Well, let's see.
So I mean, in antiquity, there was a lot of debate of, is the world discrete or continuous? So, you know, the atomists, Democritus being, I think, sort of an early one, Epicurus later, Lucretius later than that. These were all in the kind of, the world is discrete, and they had the very good idea that everything in the world with all the diversity we see might be the result of different combinations of some small number of atoms, just combined in different ways and a few different kinds of atoms.
I think Lucretius, 1st century AD., made the analogy, perhaps the world is put together out of atoms in the same kind of way that sentences are put together out of words according to the fixed laws of grammar. That was, I thought, a quite insightful idea. Anyway, so back in antiquity, there was this notion that the world might be made of discrete things. Put together in different ways. The question of whether that meant that space itself was discrete, I don't think was really addressed. I think it was more the stuff that we have.
What is it made of? There was a different tradition from people like Heraclitus, that was the everything flows tradition. Everything is really a continuous thing, and things are made out of those continuous objects. Then there were things like Zeno's Paradox, which was a confusing intermediate case about asking, if you say, well, there's Achilles and the tortoise, and they reach going this amount at this time, and so on. Without an understanding of continuous mathematics, they came in much later like by in the 1600s and so on. It wasn't really easy to unravel that kind of paradox, because that kind of paradox is asking you subdivide, and if you say, well, you can keep subdividing forever, which is the conceit of continuous mathematics and the continuous view of space, then you can resolve the paradox that way. But if it seems like you're talking about subdivisions, and you can just make a concrete subdivision at a particular level, then it still seems quite paradoxical. I think that was the story in Antiquity, and it wasn't like people were imagining space as a thing you could talk about in and of itself. I mean, Euclid, for example, in setting up geometry and writing down the way to think about geometry, his very first postulate or definition, I guess, is a point is that which has no part. In other words, he's assuming that there is a notion of an infinitesimal geometrical point. And then he goes on to say a line is that which has no breadth. So it's again, these ideas which implicitly assume kind of a continuity of space, are present in Euclid. What did Euclid think he was doing? He thought he was trying to sort of clean up what you could deduce in sort of practical terms by just drawing diagrams on pieces of slate or having rods representing different sides of a triangle and so on. He thought he was just trying to make a more perfect system that ultimately represented what is observed in the physical world. He wants it to sort of turn into philosophy in a sense, that which before had been a thing that you just did with measuring rods and so on. And that, I think, is, and so, for example, for him, the idea that two parallel lines can never intersect, was, I think, quite obvious. That was a postulate he had, his so-called fifth postulate. That was just a thing where from everyday experience, you know parallel lines just don't intersect. Now, it took until the 1800s for people to say, wait a minute, you could imagine a version of space where parallel lines can intersect, as they would do on curved spaces and so on. And then it took from the early 1800s, 1930s and so on up until Einstein in 1915 to start talking about, well, maybe physical space is actually curved in that way. But so the idea for Euclid was he's kind of reproducing sort of in an idealized form what his common experience says about ordinary physical space. And for him, it's sort of obvious that parallel lines can't cross. For him, it's obvious that there can be an infinitesimally small point that lines can have no breadth and so on. So for Euclid, he's kind of implicitly assuming there's a continuity of space. For the only sort of chink in that armor is the story of the Epicureans from Democritus on who thought of at least the stuff in the world as being made of discrete atoms. I think they still imagined that it was kind of their version of it was used was atoms and the void. So the void is that which is not part of the atoms, so to speak. And I think they kind of imagine that you could put the atoms anywhere you wanted and sort of the void was everything, which included when there wasn't stuff there. They didn't really understand gases and things like that. They certainly didn't understand vacuums. That came much later. But so for them, it was just like there's places where there's stuff and there's places where there doesn't seem to be stuff. And so all the stuff is represented as different combinations of these discrete atoms, and the places where there isn't stuff is the void. Okay. So we then go much later, I guess Descartes in the mid 1600s, is that right? Maybe a little earlier than that. I think early to mid 1600s.