Consequentia mirabilis: the dream of reduction to logic

Euclid’s Elements, read backwards, reduces complex truths to simpler ones, such as the Pythagorean Theorem to the parallelogram area theorem, and that in turn to triangle congruence. How far can this reductive process be taken, and what should be its ultimate goals? Some have advocated that the axiomatic-deductive program in mathematics is best seen in purely logical terms, but this perspective leaves some fundamental challenges unresolved.

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Here’s a way to think about one of the key ideas involved in Euclid’s proof of the Pythagorean Theorem. Picture a stack of books sitting on your desk. It has the shape of a rectangle. Let’s say you’re looking at the side with the spines of the books; they make a rectangle. Now, give the stack of books a whack with your hand. So the pile is knocked askew. The shape of the stack is now a parallelogram instead of a rectangle. But the area is the same. I mean the area of the side facing toward you, the side with the spines of the books.

It’s obviously the same area because it’s made up of the same books as before. You just moved the books around. You moved the same amount of area into a new configuration.

Also the height is the same: the height from the desk to the top of the pile. This is still equal to the sum of the thicknesses of each book.

This illustrates the geometrical theorem that the area of a parallelogram is equal to the area of a rectangle with the same base and height. This is Euclid’s Proposition 35.

This is a key ingredient in Euclid’s proof of the Pythagorean Theorem. To prove the Pythagorean Theorem we need to show that the area of the squares on the sides is equal to the area of the square on the hypothenuse. We do this by starting with one of the small squares on the sides and showing that its area can be remolded and made to fit into the big square in such a way the theorem becomes clear.

So the idea of Euclid’s proof is to transform one area into another. Its shape is transformed but the area remains the same. And the transformation he uses is basically this one with the stack of books knocked over into a parallelogram shape.

Euclid starts with a stack of books corresponding to one of the small squares. He knocks it over into a parallelogram shape. He rotates the parallelogram by 90 degrees so it’s now aligned with the big square instead. And he straightens the parallelogram back out again, just like you would straighten out a stack of books. This is how he shows the equality of areas that the Pythagorean Theorem asserts.

The book analogy is not perfect because Euclid so to speak slices his stack of books two different ways. If we want to think of his first step, transforming a square into a parallelogram, in terms of a book stack, then we must visualise the spines to go a particular way. Then when Euclid is straightening the parallelogram back out later, if we want to visualise that in terms of books, we need to picture the spines of the books differently, sitting in another direction. It’s a different stack of books, so to speak. Different but equal. If you have Euclid’s text in front of you, you can draw this into the diagram, how the books need to be oriented for each step to work, and you will see clearly that you have to change perspective halfway through. Euclid is talking about triangles instead of rectangles and parallelograms but that doesn’t matter, the principle is the same.

So we are continuing our adventure of reading Euclid backwards. We reduced the Pythagorean Theorem to a more basic proposition, the book stack proposition, 35. What does that in turn depend on? Remember that we are trying to boil everything down to its molecular components. How does Euclid prove Proposition 35? That is to say, how does he reduce this this proposition to more basic ones?

I should clarify that Euclid doesn’t do anything like this stuff with the books. I explained this theorem with this analogy to a stack of books, but certainly Euclid’s logic doesn’t depend on anything like that. That would be much too informal. The books need to be “infinitely thin” for the argument to work perfectly, and that’s a whole can of worms foundationally that Euclid certainly doesn’t want to go in to. Instead he offers a purely finitistic proof.

Euclid’s proof of Proposition 35 is very clear and satisfying. Euclid proves that one area is equal to another by adding and subtracting pieces in a clever way. So he decomposes it into a couple of puzzle pieces that fit just right with each other. Even though the two areas as wholes have entirely different shape, Euclid shows that there is a clever way of cutting the situation into puzzle pieces that are equally suited to each area.

The two areas are two parallelograms of different shape; they’re like two different languages so to speak. You would have thought that they couldn’t communicate very easily. But these puzzle pieces establish a common understanding; something that is equally natural and understandable in either language. So these puzzle pieces, this universal language, can be used to translate one area into the other.

If we think in terms of reducing the truth of the theorem to more basic facts, this means that, with the puzzle pieces, we have basically reduced the equality of the entire areas to the equality of the each corresponding puzzle piece separately. The puzzle pieces are all triangles, and the fact that corresponding ones are equal comes down to triangle congruence theorems. That is to say: Under what conditions are two triangles the same? For example, they are the same if the have side-angle-side in common.

That turns out to be the next step down if we keep reducing the Pythagorean Theorem. Like a French chef simmers a sauce to make it thicker, so we keep boiling the Pythagorean Theorem, and now we’re down to this. Triangle congruence, and some stuff about parallels as well. We have to keep reading Euclid to find out what happens if you keep cooking it.

But before we keep wilting down the Pythagorean Theorem on the Bunsen burner to see what it’s made of, let’s take a moment to reflect on this theorem about the stack of books, or the areas of parallelograms.

Proclus has an entertaining remark about this theorem in his ancient commentary on the Elements. He points out that it shows that the same area can have many different perimeters. The stack of books, if you make it more askew you will increase the perimeter while keeping the area the same. A very stretched-out parallelogram has a lot of perimeter but not a lot of area.

According to Proclus, military commanders in antiquity did not understand this, with detrimental consequences. Suppose an enemy army is advancing toward your borders. You want to know how many they are. So you send a spy in the cover of darkness at night to scout the situation. The spy sneaks up on the enemy’s night camp and stealthily walks around it, counting the number of step. He then rides back and reports this number.

So the number of steps around the camp is taken to be a measure of its size. For cities as well you could do this: How big is the city? Just walk around the city walls and count the steps. It’s so-and-so many steps big.

Of course this is a mathematical mistake, because it measures the perimeter when you really wanted to know the area. And the stack-of-books theorem shows that they are not at all the same.

Anyway, that’s just a fun story. All the propositions of Euclid have some cultural significance like this. It’s like you see sometimes the period table of chemistry and for each element they’ve added a little example of some familiar real-world thing where this element occurs. “You know kids, lithium isn’t just some weird science thing, you use it every day!” It’s in whatever, toothpaste or something. So you can do that with Euclid’s Elements as well. A little story for each theorem to lighten the mood and make things a bit more culturally relevant. But that’s just for kicks and giggles.

Let’s get back to the more scientific purpose: the systematic reduction of all geometrical knowledge to some sort of ultimate minimum foundation. We are just a few steps in to this process and it’s already starting to raise some philosophical conundrums. It was natural enough to take apart the Pythagorean Theorem into more basic results, like the one about areas of parallelograms. Then that in turn could be reduced to triangle congruence.

But this can’t go on forever. And we’re already down to such basic facts that it’s becoming very difficult to see how there could be anything “more basic” to reduce them to.

This path of reduction, it looked so natural when we set out on it. Starting from the Pythagorean Theorem, this seemed like an obvious way to go. But our clear path through the woods is now becoming darker and thornier. It’s no longer clear where to go from here. Instead of blindly forging ahead in the same direction, we need to take a step back and think about where it is we want to go. What kinds of things should the foundations of geometry be?

There are in fact a number of possible answers to this that are very different and completely incompatible with each other, yet each of them is quite plausible in their own right. Let’s have a look at some of the main ones. I mean philosophical views of the status of axioms, or starting points, in mathematics. Or what pretty much comes to the same thing: philosophical interpretation of the ultimate nature of mathematical reasoning and the source of its credibility.

Do you think mathematics is ultimately empirical, like physics? Is geometry just the science of physical space? If so, that suggests that the axioms of geometry should be the most fundamental and testable things from an empirical point of view. Geometry should start from things you can check in the field or in a lab. Measuring things with rulers, for instance. That should be the starting point of geometry if you think the certainty of geometrical reasoning ultimately derives from sensory experience and data collected from the world around you.

Or do you think mathematics is ultimately pure reason? Then the axioms don’t need to be physically testable but rather mentally fundamental. That suggests that goal of the reductive process is to boil theorems down to the most obvious or intuitively undoubtable starting points.

This divide between empiricism and pure reason is mirrored in Aristotle and Plato, one might argue. We will look into that in more depth another time.

Let’s focus now on yet another point of view: That of logic. There are two ways you can say mathematics is pure reason: One associates reason with the human mind. Intuition, aha-moments. Those are mental experiences, maybe to some extent subjective experiences. Another characterisation of pure reason is logic. This envisions the laws of reason as detached from human considerations, such as the mind and its subjective experiences. Instead it tries to give a purely objective account of reasoning.

Suppose we try to argue that mathematics is basically logic. So it’s not based on anything contaminated by humanness, such as the senses or the mind. Instead mathematical truths are simply necessary truths in some absolute sense. Their truth follow from absolute laws of reason that are some kind of abstract truths more fundamental than human experience or physical reality.

This point of view doesn’t really impose any evident restrictions on what kinds of things the axioms of mathematics should be. The starting points of mathematics do not need to be physically measurable, nor intuitively obvious, and so on. Logic does not imply such prescriptions, like the other views did.

Mathematicians just deduce consequences of definitions and axioms. Mathematics doesn’t care what the axioms are. From this point of view, mathematics doesn’t make any claim to establishing absolute truths. All of mathematics is just “if ... then ...” statements. If these axioms are true, then these theorems follow.

The axioms themselves, then, can be pretty much arbitrary for all the mathematician cares. This is a very modern view. Modern mathematicians pretty much accept this. It’s certainly a very convenient view for the mathematician. It’s a sort of abdication of responsibility.

What is a philosophy of mathematics supposed to do? What is it for? Surely it should explain the obvious facts about mathematical reasoning, such as that it somehow establishes seemingly absolute truths. When we read a proof such as Euclid’s proof of the Pythagorean Theorem or the parallelogram area theorem, the proof is so compelling. It gives us complete conviction that the theorem must be true. It’s unlike anything we ever see in other domains. There are no such absolutely compelling and irrefutable proofs in politics or ethics. Why not? What’s so special about mathematics?

History reinforces the point. Every last one of Euclid’s theorems are as true today as they were when they were written well over two thousand years ago. Every civilisation accepts these universal truths. Why does this happen only in mathematics?

A philosophy of mathematics should answer these questions. But the logic interpretation of mathematics does not. It doesn’t pinpoint any particular characteristic of geometrical reasoning that explains why it should be so unique in these regards. It doesn’t explain why the particular axioms of geometry that Euclid investigated were universally accepted in so many contexts, and turned out to be so uniquely suited to describe the physical world in all kinds of scientific advances that the Greeks had not even dream of yet.

So in this way the logic philosophy of mathematics is perhaps a kind of coward’s philosophy. It’s a non-philosophy, as far as many key questions are concerned. It just doesn’t have any kind of answer to the major questions that other philosophies of mathematics sees it as their duty to address.

There’s a famous essay called “The unreasonable effectiveness of mathematics in the natural sciences.” Famous physicist Eugene Wigner said this in 1960. Everybody cites it all the time.

But ask yourself: Why did no one say this until 1960? Did the effectiveness of mathematics somehow become unreasonable only then? Of course not. The effectiveness of mathematics in the natural sciences had been around forever. Including the effectiveness of ideas that were first developed for purely mathematical reasons but later proved to have hugely important and completely unforeseen scientific applications. For instance, the Greeks studies ellipses in great mathematical detail, and then two thousand years later it turned out, completely unexpectedly, that planetary orbits are ellipses. So this purely geometric topic became hugely important in science, which no one had predicted.

Why didn’t people say then: the effectiveness of mathematics is unreasonable? Why would it take all the way to 1960 before anyone drew this obvious conclusion?

I’ll tell you why. Because the conclusion that the effectiveness of mathematics is unreasonable only follows if one assumes the logic interpretation of mathematics. If mathematics is nothing but logical inferences from arbitrary axioms, then sure enough it’s a complete mystery, it’s completely unreasonable that mathematics can work so well.

But what people used to conclude from this is that it is the logic conception of mathematics that must be unreasonable. It is unreasonable to think that mathematics is nothing but logical deductions. Because that completely fails to explain so much of what we know about mathematics.

In 1960 the logic conception of mathematics had become the modern dogma that it remain to this day. It had become so ingrained in the mathematical psyche that mathematicians could no longer even conceive of rejecting it. Then they had no choice but to declare the effectiveness of mathematics in physics to be unreasonable. That’s why Wigner’s famous phrase is from 1960 and not 450 BC.

It’s not a fact that effectiveness of mathematics is unreasonable. Rather, one of two things is unreasonable: either the effectiveness of mathematics is unreasonable, or the conception of mathematics as nothing but logic is unreasonable.

For thousands of years people preferred to conclude from this that there must be more to mathematics than just logic. Euclid is not just “the axiomatic-deductive method.” This can’t be the whole picture. The axioms must be somehow more than arbitrary. What makes the axioms true? Logic itself doesn’t care and cannot help us with this question. So we need something more than logic in our philosophy of mathematics.

So I claim that only in very modern times did the logic conception become the norm. Maybe in some future episode I will discuss what circumstances made that come about. The important thing for our present purposes, as we read Euclid, is to understand that with the reduction process that we have begun, that consists of breaking down theorems into smaller and smaller pieces, the end pieces, the ultimate rock-bottom pieces, need to have some sort of claim to credibility. They cannot simply be whatever you’re left with when you keep reducing and reducing.

Or can they? I say everyone rejected that view, but I could play devil’s advocate. Listen for example to this fragment from Eudemus’ Physics: “As for the principles they talk about, mathematicians do not attempt to demonstrate them, they even claim that it is not their business to consider them, but, having reached agreement about them, they prove what follows from them.”

This is a bit of a disturbing quote, in my opinion. It seems to almost assert that logic view that I said was regarded as unacceptable at that time. Mathematicians only prove what follows from axioms, and they claim that “it is not their business” to worry about the status or truth of those axioms. Sounds strangely modern, just the view I assigned to the 20th century.

I think that’s not really what the quote says for various reasons. In part what Eudemus is saying is that the justification of the “principles” (that is to say the axioms) shouldn’t be regarded as part of mathematics but rather part of some other field, some more philosophical domain. But whatever, that’s just putting labels on things. That still means that the axioms are to be justified some way. So they are not arbitrary. The justification is “philosophy” rather than “mathematics”—sure, whatever, call it what you want, but it’s in any case very different from not justifying or being concerned with the nature of the axioms at all.

The quote also said, if you noticed, that the mathematicians don’t care about the axioms, “having reached agreement about them.” What does that entail? On what basis did mathematicians reach such an “agreement”? This opens the door for all kinds of considerations of the status and nature of the axioms within mathematics, even according to this quote, the devil’s advocate quote.

So I think it’s safe to say that the logic view by itself was not satisfactory. The starting points, or axioms, of mathematics need to have some kind of justification.

In fact, there is one way in which logic itself can provide such a justification. So the problem we need to solve is this. We started with the Pythagorean Theorem, we reduced it to more basic statements, then those to more basic ones, and so on. Where do we stop this process?

Do we stop when we just don’t see how to go any further? This is what I just criticised as untenable. Because this would mean declaring whatever we’re left with to be axioms, without convincing criteria of justification for which kind of things should be allowed to be axioms and which not. The axioms can’t just be arbitrary because then we can’t explain the successes of mathematics.

One hope of some logicians has been that everything could be reduced to definitions. There are no axioms! Everything is at bottom just definitions. The meaning of words. Mathematics is about drawing out consequences contained in the definitions of concepts, without any assumptions being made.

That would be great for the logician and some people have tried to fit geometry into such a mold. But it doesn’t work. Geometry needs assumptions, genuine axioms. You can’t get away with only definitions. You can’t reduce mathematics to a purely linguistic game. And besides, even if you could, what would be the guarantee that the definition corresponded to anything? That the entities defined actually exist? And that the definitions are not self-contradictory or inconsistent? Definitions alone cannot carry this burden of justification. You need something more.

But there’s one more ace up the logician’s sleeve, and it’s a pretty clever one. There are statements that are logically self-justifying. Statements such that, if you try to deny them, you have actually committed yourself to accepting them.

An example is the famous statement by Descartes: I think, therefore I am. How could you deny such a thing? What would you say if you wanted to deny it? “No, I don’t think that.” Or: “I think that’s wrong.” As you can hear, you walked right into the trap. By trying to deny that you are a thinking being, you made statements that actually presuppose that you are thinking being. The denial is self-defeating. You can’t deny the statement without actually implicitly conceding it.

Such statements are justified by “consequentia mirabilis,” as it’s called.

There’s an argument of this form already in an Aristotelian fragment. Aristotle uses it to prove the proposition: We ought to philosophise. Try to deny it. So you say: No, we should not philosophise. Well, in that case, it would be important to reach the conclusion that we should not philosophise. Reasoning our way to this conclusion would spare us from the mistake of philosophising. Then we could do more important things with our time instead of philosophising.

But now we are caught in a trap again. We wanted to establish that we shouldn’t philosophise, but in trying to argue this we actually committed ourselves to the position that we should philosophise, namely we should philosophise in order to establish the conclusion that we shouldn’t philosophise. So once again the attempted rejection of the proposition actually implies acceptance of the proposition.

Could it be that all the axioms of mathematics could be of this type? That would be a logician’s dream. That would be a great way of justifying ending the chain of reductions of theorems to lower and lower constituent parts. We have to keep reducing until you’re left with nothing but logically self-justifying statements. Consequentia mirabilis axioms only, which must be accepted as true because it is logically incoherent to try to deny them.

This view had its adherents. Clavius was fond of the consequentia mirabilis. Clavius was influential in discussions of Euclid around 1600; he was the editor of the standard Latin version of Euclid that everybody used. Even Saccheri, who did some very sophisticated work on the foundations of geometry in the 18th century, was keen on trying to reduce the foundations of geometry to consequentia mirabilis.

So this idea was clearly seen as very attractive. People really tried to make it work. But ultimately it failed. It was an approach based more on what the logician wanted than on what mathematics is really like and how mathematics wants to be understood.

So altogether, the reduction of mathematics to logic is an idea that has had great appeal to many. Several times in history, a complete reduction of mathematics to logic has seemed within reach, only for the quest to end in bitter disappointment. This is also what happened with Frege and Russell, Hilbert and Gödel, and so on, centuries later.

Bertrand Russell put it in interesting terms. Here’s what he says in his autobiography: “I wanted certainty in the kind of way in which people want religious faith.” He’s talking about his early career, around 1900. At this time he worked on an enormously ambitious project to reduce all of mathematics to logic. It didn’t work. As Russell himself says: “After some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.”

Russell’s case is quite typical, one might argue. Others have had the same experience when they have tried to achieve the same goal. It’s a great temptation: one logic to rule them all; “my precious.” Many have been seduced by that idea, and spent twenty years obsessed with it only to fail, as Russell did.

Let’s look at the most famous of the problems Russell ran in to: the so-called Russell’s Paradox. A popularised version of Russell’s Paradox goes like this. A barber shaves everyone who do not shave themselves. Who shaves the barber? There is no coherent answer. The barber cannot shave himself because he only shaves those who do not shave themselves. But he also could not not shave himself. Because if he didn’t shave himself he would by definition be one of the people he does shave, which is everybody who do not shave themselves. So either way leads to a contradiction.

Mathematicians unknowingly allowed this type of paradox to enter their logical systems. This stuff about the barber is just a translation into everyday terms of something that first occurred within mathematics itself.

Russell thought this problem was fixable. But others thought it was a comeuppance for logic that was both deserved and bound to happen. Consider for example the reply by Brouwer, an influential but eccentric mathematician in the early 20th century. Here’s what he says: “Exactly because Russell’s logic is no more than a linguistic system, there is no reason why no contradictions would appear.”

That is to say, since logic is divorced from meaning, divorced from the real world, why wouldn’t it be inconsistent and self-contradictory? History shows that inconsistencies can very easily creep into formal axiomatic systems, against the best efforts of even top mathematicians devoted specifically to building rigorous and coherent foundations. A long list of leading logicians have published systems of logic which turned out to be inconsistent.

According to Brouwer: “The language of Euclidean geometry is reliable only because the mathematical systems and relations, which are symbolized by the words of that language as conventional signs, have been constructed beforehand independently of that language.”

That is to say, it is precisely because it is not merely logic that Euclidean geometry is so reliable. It is anchored in the real world, and the physical world has a much better track record of being consistent than the thought-constructs of logicians.

Emil Post was another rebel at that time who likewise called for “a reversal of the entire axiomatic trend of the late 19th and early 20th centuries, with a return to meaning,” as he put it.

Logic had gone too far. Some formalisation and logic are powerful tools in mathematics. But you can take it so far that mathematical theories lose all bond with reality and meaning. Then there is no grounding anymore to protect you from contradiction and inconsistency.

Logic is “the hygiene which the mathematician practices to keep his ideas healthy and strong,” said Hermann Weyl, another contemporary of these guys. But, like hygiene, you can overdo it. Some hygiene is much better than none, of course, but obsessive hygiene can undermine the natural state of the body and the immune system. Maybe logic is like that. It’s like cleaning everything away with bleach all the time. It’s good to clean, but if you overdo it you eventually clean away the very thing you were trying to protect.

There were big debates about such questions in the early 20th century; the people I quoted were all part of those heated debates about logic. But that’s a story for another day. For our purposes, we are interested specifically in logic-centric attempts at interpreting Euclid, and accounting for the success of Greek geometry. Indeed, such logic-centered interpretations have been sought eagerly. They are very agreeable for some purposes; they have an almost religious appeal, as Russell said. But ultimately there are severe limitations inherent in such views, which have meant that most people from antiquity to early modern times have felt that some additional ingredient, beyond mere logic, is needed for a successful philosophy of mathematics.

And as we read Euclid backwards, the closer we get to the beginning, the more essential it becomes for us to make up our minds about our philosophy of mathematics. Any moment now we have reached all the way down to the axioms and then push comes to shove. We’re going to have to take a stand and say: this is why we stop at these particular axioms and why you should believe them. Let’s keep reading Euclid and see how we can answer this challenge.