This Simple Question Tests How Well You Study Math

追加: 2026-05-31

[00:00:00]Let's say I ask you to define something mathematics. An object that is so simple that it almost feels like a trick question. What is a point in space? It sounds simple and you may think that you know what a point is, but how can you actually define it? The way you answer this question will show whether you know the best way to learn mathematics or not. This is a question in great point career raised. And he says that first of all, you should ask yourself, is it even possible to imagine a point in space?

[00:00:27]If you said yes, most likely what you are really imagining is a black spot made with a pen on a white paper.

[00:00:34]Or a white spot with chalk on a blackboard.

[00:00:37]What you are actually representing in your mind is your personal impression of a point.

[00:00:42]In any case, define it as you like, but here's something you have to remember.

[00:00:46]It is not enough to define one point.

[00:00:48]You also need to know how to distinguish it from other points. The question then becomes, the point that you thought about a minute ago is the same as the one you're thinking right now.

[00:00:59]Thinking in physical terms, how do we know that the point occupied by the object A at the instant alpha is the same as the point occupied by the object B at the instant beta?

[00:01:12]Let's say you are seated in a room and you place an object on your table and wait for 1 second.

[00:01:20]You may be tempted to say that the point A, which the object occupied a second ago, and the point B, in which it finds itself, are identical.

[00:01:30]But no, between A and B are 18 and 1/2 miles because the object continued to move with the motion of the Earth.

[00:01:39]So, here's the thing. Same place depends on your frame of reference. Relative to your room, it is still, but relative to the sun, for example, it is moving. Now, notice what just happened. We started with one of the most simple objects in mathematics, a point. But as soon as we started to ask, how can we define it?"

[00:01:58]Our intuition started to blur. And this is where Poincaré's point begins, which he widely discussed in his book The Value of Science, where he said that intuition cannot give us rigor.

[00:02:10]This has been proven time after time in mathematics, and real-world examples are not hard to find, like the Dirichlet's principle, which is something that many theorems in mathematical physics rely on.

[00:02:21]A certain energy functional defined on a class of functions U was known to be bounded below, meaning it had an infimum.

[00:02:30]Earlier mathematicians assumed that this automatically meant that there existed some function U0 that actually attained the lowest energy, so that E of U0 is the minimum of E.

[00:02:41]The flaw is that an infimum need not to be a minimum. Functions may get closer and closer to the lower bound without any admissible function U ever actually reaching it.

[00:02:53]And as a consequence, intuition cannot give us certainty, and that's why mathematics had to rely more and more on rigor, especially compared to how mathematicians used to think in the past, more geometrically. So, the question is, is there even any need for intuition? At this stage, the lesson is obvious, right? Intuition is just useless. We don't need it anymore. It is something dangerous, and rigor is here to save us. That's why we should just leave intuition behind. But not so fast.

[00:03:22]Remember our point analogy. Logic can reason about points, but once we already have a grasp of it. But would logic alone have ever told us to even invent the notion of a point in the first place? Poincaré's answer to this question is no. But let's have Sophia tell us about it. Logic or rigor alone would only lead us to tautologies, which is a word that basically means a statement that only repeats ideas.

[00:03:47]What Poincaré is saying is that pure logic can only make explicit what is already implicit in the assumptions, definitions, or axioms. So, it can preserve and clarify the truth, but it can't by itself create genuinely new mathematical knowledge.

[00:04:03]Poincaré says that to make arithmetic, as to make geometry, or to make any science, something else than pure logic is necessary.

[00:04:11]This something else, it's our intuition.

[00:04:14]But, the thing is intuition is a really, really, really broad word. And sometimes people tend to use it way too loosely when they mean things like relying on your geometric sense, or relying on how you perceive the environment, and things like that. To understand what I mean, take a look at these four axioms. And the point is not for you to understand each of them deeply, but to see how intuition relates to every one of them.

[00:04:39]First, two quantities equal to a third are equal to one another.

[00:04:44]Second, if a theorem is true for the first term, and assuming that it is also true for the nth term, we can prove that it is also true for term n + 1, then it will be true of all whole numbers n.

[00:04:57]If in a straight line, the point C is between A and B, and the point D between A and C, then the point D will be between A and B.

[00:05:06]And finally, given a line and a point not on that line, there is at most one line through the point that is parallel to the original line.

[00:05:16]People often say that these axioms were formed thanks to our intuition because they're really easy for us to grasp.

[00:05:23]They're kind of obvious. But, Poincaré says that throwing all of them under the word intuition is way too vague because intuition is being used in very different ways here.

[00:05:33]The first axiom is just a statement of a rule of formal logic. It doesn't depend on imagining some kind of space or constructing numbers. It's just basically a logical rule.

[00:05:43]The second is an a priori judgment, which means that we can't learn this by checking real physical examples, only through thought. So, for example, if you take an infinite staircase, you know two things. That first, you can stand on the first step, and second, whenever you can stand on one step, you can climb to the next step. This is how you know you can reach every single step.

[00:06:03]What Poincaré is saying is that this conclusion is not learned from experience. You're not going to test every single stair. But what it does do is express your intuition of an endless ordered sequence.

[00:06:15]The third heavily relies on your imagination. It needs your spatial reasoning to have you understand it.

[00:06:22]And the fourth is actually a definition in disguise.

[00:06:25]It looks like a fact about space, but it's actually convention that tells us what kind of geometry we're using. So, Euclidean geometry. Because in non-Euclidean geometries, this statement is actually false.

[00:06:38]So, that's why intuition is not just one simple thing. Different axioms are called intuitive for completely different reasons. So, intuition is not just relying on our senses. And this becomes especially clear when we take a look at something called Poncelet's principle of continuity. Poncelet was a French mathematician who was known for his really intuitive style of geometry.

[00:06:59]But he's also someone who said that if a mathematical statement is true when the quantities are real, we should often continue to treat it as true when those quantities become imaginary.

[00:07:09]So, for example, a hyperbola has real asymptotes. These are actual lines you can draw, which the hyperbola approaches.

[00:07:16]But an ellipse does not have ordinary visible asymptotes.

[00:07:20]Even though in algebraic or projective geometry, we can formally talk about its imaginary asymptotes.

[00:07:28]This is just to emphasize the fact that intuitive doesn't necessarily mean easy to picture. He had an intuition for something that he did not prove mathematically, but he kind of had a feel for it. So, now that we've settled what exactly Poincaré means by intuition, we can't really say that he's proven that it's something that's really really necessary for mathematical thinking. Maybe it's useful for a student who's learning something for the very first time. But for someone who's already mastered the subject, is it really a useful thing? And is it even necessary? Well, let's take a proof and instead of seeing it as one flowing idea, cut it up into a bunch of individual logical steps.

[00:08:08]If we checked every little step and confirmed that each one is valid, would that mean that now we understand the proof as a whole?

[00:08:15]No, because checking correctness is not the same as getting the meaning.

[00:08:19]Well, what if we were to memorize each individual step and then reproduce it again following the order? Would we understand the proof then?

[00:08:27]Again, no.

[00:08:29]You can recite a proof perfectly, but still have no idea what is the big idea behind the proof. Formal logic can break a proof into valid steps, but what it won't show us is why those steps belong together. And plus, when we're dealing with a problem, mathematics gives us countless valid methods to solve it. And if we pick one of these paths and follow through the rigorous procedure correctly, sticking to correct calculations, we're not going to run into contradictions or invalid steps.

[00:08:55]So, we'll have a valid solution.

[00:08:58]But does that mean that the road we chose to solve it was useful?

[00:09:02]How do you know which path to choose?

[00:09:04]The answer is intuition. So, the ability to have the end in mind and knowing where you're taking a specific direction.

[00:09:11]Intuition helps both the person who creates the proof and to the student who's trying to understand it.

[00:09:17]>> If you're going to remember just one thing from this video, remember that when mathematics becomes rigorous, it forgets its historical origin. So, we can see how we can get answers to questions, but we forget why we had this question in the first place. We can give the most precise definition of a point, but if we don't even know what it is for and why we're doing it, the whole thing becomes pointless.

[00:09:39]Ironically. And this is something that, at least according to Poincaré, shows that logic alone is not enough. That's why we believe that starting with intuition is the best approach to study mathematics. Even though later on, rigor must complement this intuition that you built. So, if you're actually serious about learning and expanding your knowledge using an intuitive approach, check out our catalog of PDFs.

[00:10:03]Each of them comes with a YouTube video.

[00:10:05]And they are built on intuition, concrete examples, rigorous explanations, and only then exercise with detailed solutions.

[00:10:13]Check them out in the description below.

[00:10:17]Poincaré concludes, "Thus, logic and intuition have each their necessary role. Each is indispensable. Logic, which alone can give certainty, is the instrument of demonstration. Intuition is the instrument of invention."

[00:10:33]Or in other words, logic solidifies our steps, but intuition gives us direction.

[00:10:38]Now, if you want to see what Poincaré has to say about two different kinds of mathematical minds, click on this video right here.

[00:10:46]I'll see you there.