Hey, everyone, welcome to Nontrivial. I'm your host, Sean mcclure. In this episode, I argue that intuition is more rigorous than precisely defined arguments such as those constructed using mathematics. I talk about the cost of precision in real world situations, offer an approach to formalize the notion of intuition and suggests that when it comes to establishing the connection between our ideas and reality, we should always let intuition lead. Let's get started.
So we all have world views based on what we think about the world, more specifically how we think the world works, right? We go through life, we have experiences and then we, we pick up on, you know, kind of signals and ideas and, and we have relationships with people and this, this forms kind of these isolated concepts that we have in our mind.
Uh these kind of definitions that we create about things, whether it's capitalism or socialism, anything in the political sphere, anything about human relations, you know, religion, whatever it is. Uh And then, and then we kind of piece those together into, into kind of these causal stories, right? These causal structures. Well, this is a and this is B and this is the way A and B interact. And this is why C happens.
And we come up with all these stories in our, in our mind and, and these are our world views, right? And uh and we would use those to explain things if someone asked us about, you know, why did you take the job? You did? Why do you do what you do for a living? Why do you have this political leaning? Why do you believe what you believe? Then we would offer those explanations? But of course, those explanations are largely emotional.
And what I mean by that is it's not like we're being super specific or super, you know, scientific or analytical in those explanations. We're just saying, well, this is how I feel about this and this is how I feel about that. And I think this and that come together in this way. And therefore, here's my story about who I am and why I believe what I believe and why I take the actions I do. But sometimes we need to be uh more rigorous with our explanations, right?
By rigorous, I mean, kind of less hand wavy, less emotional, a little more analytical, more logical, more specific. You know, here's specifically what I mean. Uh less open to, to, to being misinterpreted, right?
A rigorous explanation of something is, is, is very exact, is, here's exactly what I mean for why this is my explanation for what's happening and you can imagine in science and engineering, obviously, we try to be more rigorous than just, you know, someone's explanation of their world view, right? What is the real underlying mechanism at play here? How do you, how do you be more exact with that?
And the way that we try to be rigorous with our explanations in things like science and engineering or any time we're trying to be rigorous is via precision. precision is when we are kind of removing kind of the messy details of life, tho those seemingly superfluous kind of externalities, stripping all that away and, and just bringing our definitions down to something that seems irrefutable, right? You know, it could be facts that we piece together logically.
Uh it could be kind of statistics that we gather uh definitions that we come up with that are much less prone to misinterpretation. We we'd be very precise with what we're, what we're speaking about. And that allows our explanations to presumably as is commonly thought, be more rigorous, right? Because there's less chance that it's wishy washy, it's not all emotional, it's not kind of hand wavy. Things are very precisely define, but of course, even lies can be told with things like facts.
I mean, I talked about this in my logic episode a couple of seasons ago. Uh we can take facts that are in and of themselves true and then rearrange those truths into a story and that story could be totally false, right? Because we can abuse our human dependence on causality, meaning we want things to add up, right? We want, we, we can only really understand things as a story, right?
This is A and this is B and then A and B interact to produce C we think about the the kind of the causal structure behind things. That's how things make sense in our lives. And we can kind of abuse that and this happens, you know, in the news media, you know, some people do this where you know, and and conspiracy theories can kind of work like this, right?
Where you take a bunch of things that by themselves are true and then you kind of mush them together and then it tells a story and it does tell a story and it can sound very convincing, but the story could be uh totally not true, right? Totally untrue just because of the way you rearrange those facts. So facts in and of themselves precisely define constructs in and of themselves does not mean truth, right?
And this is why things like mathematics and rigor have gone hand in hand for centuries because mathematics does not allow all stories to be told. You can only take your precisely defined ideas and concepts and and arrange them in certain ways, right? We know that there are operations in mathematics, right? Like addition, subtraction, multiplication, division.
And there are ways that this can happen in math, like the machinery, the internal consistency of the mathematics language only allows so many stories to be told in certain ways, still a lot of stories. But they, they have to follow these sets of rules. And this is why generally in society that if, if something, if an explanation is given uh mathematically, it's, it's, it's kind of accepted as being a more rigorous explanation of the thing.
You can, you can imagine something like in biology, which is maybe usually not that mathematical, maybe they have a theory about some organism. And, and here is where cellular yada yada is going on. And then they say, OK, now we're going to put forward a mathematical description of what we're talking about.
Well, now this looks very rigorous right now, things are being defined precisely uh there's less chance to misinterpret what's being explained and, and, and this would be accepted by and large as a rigorous explanation of this particular, you know, phenomenon. Let's say when we convert our ideas into mathematical form, uh you know, we create the most precise symbolic version of the mechanisms we believe drive the things we observe and measure. But now I wanna talk about the cost of precision.
So mathematics is about creating symbols, right, that represent some notion in abstract terms, symbols like the equal sign, right? So symbols for equal or greater than or pi uh or a function like F FX, you know, these are labels that mean something specific uh think about the uh you know, the idea of a function, let's say now we can say a function is something that converts inputs out puts like a box that transforms information by doing something in the middle.
But such a hand wavy definition would not suffice in mathematics, right? Because there can be no room for misinterpreting what we mean when we say function to be, you know, rigorous, we have to define function by saying something, you know, quite precise, like uh you know, a function assigns elements from a set X to a set Y such that each element of X exactly maps to one element of Y or something like this, right? And this is what a function F of X would mean precisely, right, that symbol.
And with that kind of clear cut definition in hand, we could, you know, we're free to use that symbol along with other precisely defined symbols and combine them into the kind of indisputable stories that rigo depends on. But precision is not the same as accuracy, right? Accuracy is about how close our measurement is to the true value. Whereas precision refers to how refined our measurement is, and we can kind of visualize this uh a common way to visualize this would be with a target.
So you think of like if you're shooting arrows at a target and in the center of that target, you have a bull's eye. And if you are close to the bull's eye of that target, you're accurate. Ok. So, so the bull's eye of the target represents the truth, right? Whatever the real, real truth of the situation is, if you hit that bull's eye, then you're, you're accurate, right? You're very close to the truth.
And if you're far from it, then you're obviously not as close to the truth and precision would be if you shot a bunch of arrows and all those arrows were kind of tightly packed together, then those would be, uh, you know, a precise shot. They're very similar to each other. They're tightly packed. Um, but we can see that, you know, precision and accuracy are not the same thing because we could have a bunch of tightly packed arrows.
So a bunch of precision, but those tightly packed arrows could be far away from the bull's eye, right? They could be sitting in, let's like, like the top left level of the target, let's say so, uh, just because you're precise, definitely that has nothing to do with you being accurate, right.
And, and so we could actually say that there's two main ways that we could have low accuracy either by having low precision where our values are so far apart, they cannot possibly land on the bull's eye, right. So you're shooting a bunch of arrows and they're so spread out that you, uh, you would have to say, well, that's not good accuracy because they're so spread out, there's no way you could really land on the bull's eye, right? Uh a regular number of times.
Um And uh we could also say we could have low accuracy by our precise measurements deviating as a group from the true value, which is the one I already said, where the with a bunch of arrows are together, but they're not hitting the center, they're not hitting the bull's eye of that target. So, so why am I saying this? Well, accurate and accuracy and precision are distinct concepts? Ok. Um And, and so just keep that in mind that these are not the same.
So just because you're being precise, doesn't mean you're being accurate, right? We can already see that they're distinct concepts. But if that was the whole story, then, then we would think, well, couldn't we achieve both high precision and high accuracy like that seems conceivable, right? Um Which is true like why not just have a bunch of arrows that are tightly packed but that tightly packed bunch of arrows are sitting in the bull's eye.
So you're both highly precise and highly accurate, right? That seems conceivable. But what's not communicated in that scenario is the cost of precision when it comes to real world situations. So this is, is particularly true uh under complexity, right? Which is, you know, again, 99% of everything, right? Real world situations, which are, you know, causally opaque, very complex.
A lot of uncertainty there, it turns out that there is a cost to the precision and, and this is more of the story that I want to get into. So coming back to mathematics with the symbols used in mathematics, we have to understand that they have been stripped of context in order to make them strictly defined.
OK, the context of a situation which all those kind of details, the ins and outs that come into something, you know, it's, it's like if I say a word and then you think about all the different words that accompany that word, that context is what really gives it that meaning in order to be really, really precise, I kind of have to strip all those other things away, isolate it, kind of sterilize it, compartmentalize the thing I'm talking about.
Take away that context and be very, very precise with it so that, you know, it's not vague, it can't be misinterpreted, it's strictly defined, but to strictly define something by definition means I had to remove all that other stuff. I had to remove the context, the surrounding kind of material. If you will, that defines the situation, right? Consider a dictionary, a dictionary does not contain the meaning of words, only their definitions, right?
Meaning and definition is not the same thing a dictionary will define will use words to define, you know, kind of specifically what a word uh is, is supposed to be defined as, right? The definition of that word. But the meaning of the word can only come by thinking about where that word is. If it were, you know how that word would be embedded in a paragraph or someone's speech, for example. Right. It's the words that surround that word of interest that give that word its meaning.
OK. And of course, that's the context. The precise symbols in mathematics only get their meaning by virtue of being embedded in some real life situation. You can imagine the people coming up with these mathematical symbols in the first place, they're likely toiling away in some real world situation, trying to understand the patterns and then landing on some salient point about the situation and then trying to capture that as a concept.
But in order to really capture that as a concept, they had to strip all that other context away and present it as just this singular symbol, right? To present that singular symbol to someone is is is by definition to not carry the context with it, right? Only the person who came up with a symbol had the real life context that made it in the first place, right? To be precise is to by definition, remove the context that allowed you to arrive at that symbol in the first place.
So the cost of precision is that we must remove the very thing that makes something realistic, which is context. And so because of that, I'm going to argue that intuition is actually more rigorous than mathematics. Now, this is not me saying mathematics is not rigorous. This is not to say that to, to practice kind of the art of precision does not bring a certain level of rigor to something I think it does. But it's intuition that is truly rigorous.
If we think about how we're defining rigorous, which I want to argue is, you know, something's connection to the true accurate value, right? The underlying reality. OK. So what differentiates intuition from other forms of knowledge acquisition is its lack of conscious reasoning, right? It's lack of conscious reasoning. So when we, when we say we're being intuitive about something, it's more like an emotion, a feeling, we feel that something is true about this situation.
Maybe it's a feeling you have about a person about a job, right? About some uh situation where maybe the violence is all of a sudden increasing and you get some intuition, you know, do you, do you, do you stay or do you run or whatever it is, right? We have lots of intuitions all the time throughout, throughout our lives and, and and on any given day, but the intuition is a lack of conscious reasoning.
We're not being analytical, we're not, you know, weighing the pros and cons, we're not thinking about how the details add up. We're kind of getting this high level sense of, of the essence of a situation if you will. But we're brought up in society to view intuition as kind of being a weak or less rigorous notion of truth, right? As though it's, you know, maybe the starting point, maybe the catalyst to something to, to, to some deeper kind of analysis that will come later.
But, you know, surely you can't rest the argument on intuition. I mean, it can't be, you know, the ultimate explanation of something. It's, it's almost seen as kind of the opposite of rigor if you will. But remember that when we talk about precisely defined things like the symbols used in mathematics that you know, are facts that you might gather that these are true only in the narrowest sense. Right? Again, maybe you can make a lie using nothing but facts.
The lack of articulation associated with intuition is not less true. In fact, I would say it's quite the opposite. What should be considered truly rigorous is how connected our measurement model or explanation is to reality. While mathematics restricts the kinds of stories we can tell this does nothing to guarantee such restrictions map to reality.
I mean, again, you can be as precise as you want, but you're choosing to be precise because of the benefits that precision give you such as it's harder to misinterpret what I'm saying. But that says absolutely nothing about how well connected you are to reality. Mathematics cannot be the most rigorous of depictions because it lacks almost everything that made something recognizable. Keep in mind that intuition has been with us for millions of years of evolution, right?
Evolution has endowed us with this remarkable ability to connect our minds to our environment, to sense what is there to make decisions based on what we feel about a situation, right? So intuition um is is given to us by evolution for obviously non random reasons. It's there for a reason and it's arguably the primary way in which we navigate our complex world.
So the connection between our minds and the, the, you know, the truth of a situation, which is what we do in science, what we do in engineering, what we do every day and, and what we're attempting to do when we make a rigorous explanation of something intuition is what has that, that connective tissue between our minds and reality. Our interpretation of intuitions could often not be correct, but the connection is still residing within the intuition.
OK. So let's step back and try to formalize intuition a little bit. Um Because even if you kind of accept what I'm saying and say, yeah, and I can see why intuition, even just for evolutionary reasons must be connected to, you know, uh the reality of a situation in ways that precision can't be because of the context that gets removed through precision. Um You know, maybe it still seems a little bit hand wavy or maybe it still seems it's hard to anchor on something, right.
Because without a story, we fail to comprehend much of anything. And a story is really constructed from, you know, fairly precise facts about things that we combine into, you know, this kind of causal structure, right? We take facts about a situation, we combine them, we create a narrative out of it, uh out of those facts, out of that, out of those connections. And that's how we understand our world. So without a story, we fail to really comprehend much of anything.
So part of thinking about intuition and why it should be perhaps considered the most rigorous of all things. Um We still need a story to do that. We need to kind of formalize that, right? We, we kind of need to be precise about it precise about intuition. Now, isn't it hypocritical? To first argue as I have about the cost of precision relative to intuition and then suggest we could benefit from a more precise definition of intuition seems kind of hypocritical.
Well, such circularity are avoided as long as we can formalize our ideas without losing context because that's the whole point that I've been arguing is that if you go to be precise, you have to strip away the context. And uh and that's the cost is, you know, is there a way to kind of be precise about something without losing context? Well, it kind of seems impossible.
But one of the best examples of this approach is how uh logician Kurt Goodell or Girdle was able to showcase the limitations of mathematics by using meta mathematics. OK. So the idea here is, you know, Goodell Gle, he came along and he wanted to show that, you know, mathematics probably couldn't prove all things, there are limitations to mathematics. And he wanted to kind of show this mathematically, right?
Because again, that would be the rigorous thing to do or at least he wanted to be precise in his arguments. But it wouldn't make sense to use mathematics to show the limitations of mathematics, right? That would be circular and be kind of hypocritical. It doesn't really make sense. I mean, how could you use the language that you're trying to show the limitations of by using the language itself? Right?
Um but if you could step outside mathematics and use something like a meta mathematics, then you would be able to do this because you're not using mathematics directly. What is mama, you know what is meta mathematics? Well, think about metadata, right? If I say something is data, we all know what data is. It's kind of a way of logging the information about things, right? But metadata is data about the data, right? It's about it.
So maybe like how many rows are in your, in your table, how many columns, how many missing values, right? It's data about the data. OK? So when you step outside the system and you and you and you speak about it from a meta perspective that allows you to avoid some of that. Well, all of the circularity really because you're not contradicting yourself, right? Uh Maybe a more down to earth example that people could relate to.
Imagine someone trying to, you know, maybe argue for uh the existence of God. And then all of their arguments are based on the Bible. Well, that's circular, right? Because you, you already believe in the Bible, but the person you're talking to might not. So you can't use the Bible to argue for the existence of God that would be circular. You'd have to step outside the the theology and kind of argue for the existence of God maybe on purely you know, logical terms or something like this.
So the point is is that when you go meta you, you step outside the system and you try to be precise with your, your your meta mechanics, if you will about what it is, you're trying to argue. So one way you could do this is with something called C theory and category theory.
Uh If you look at it, it looks kind of mathematical, although it's got more pictures, it's not math, it's actually about math because meta mathematics was or specifically category theory was invented to basically show the connections between different areas of mathematics. It would help essentially facilitate conversation between mathematicians, right? So, so category theory is not math, it's about math, right? Just just just like metadata is about data, right?
If you want to understand how different pieces of data were related to each other, you would look at the metadata, right? And then you'd kind of see, oh this table has the same amount of rows as this table and things like that anyway. But how can we bring category theory to the problem of formalizing intuition? Well, to start, I would say that I would argue that intuition is in fact analogy or more precisely analogy making.
OK. Uh If you think about what you're doing, when you are intuiting something, when you have intuition about a situation, you are basically making an analogy between some category that you already have in your mind and some category that is apparent in a situation that arises. So somebody walks into your office and you're gonna hire them for a job and you kind of have an intuition about that individual. Well, you have, you know, a category in your mind about the job itself, right?
Or maybe a few categories about what constitutes the requirement for the job. And then the person as you talk to them, you start to kind of categorize that person based on, you know, their personality and their experience and what they're talking about. And you kind of make a connection between those two, right?
Because that's the purpose, that's the problem you're trying to solve is to hire this person for the job, you know, and, and again, there's gonna be some analysis and deduction there, but largely it's intuition, right? You either feel this is the person for the job or not. So there's a connective tissue between the categories that already kind of preexist in the mind and the categories that you're making about this person as you're talking to them.
And that connection between categories is like an analogy, right? Any time we make an analogy, we find essentially a connective tissue between things that seem superficially different, right? What does this person I've never met before have to do with the job I'm hiring for initially nothing.
But then as you talk to them and you get that context of the situation, you kind of make analogies between maybe their skills that they have and the skills that are required, maybe those skills seem, you know, to have nothing to do with it at the beginning.
But as you talk to them, you realize actually that would be a transferrable skill because what you did in this organization is a is actually analogous to maybe some of the stuff we've done over here, even though maybe the industries were different, right? Whatever it is. So I would argue that intuition is really analogy making. You're just doing a bunch of analogy making. And I would argue analogy making is very much kind of at the root of cognition, right?
As some other people have argued, well, if we accept that intuition is rooted in analogy then we're actually primed to be more formal in our treatment of intuition, which is what I said, we're going to try to do here because analogies can be expressed rigorously in terms of categorical structure. And that's kind of what I was talking about already.
You have a category in your mind, you have a category of the situation where you can use this meta mathematics called category theory to basically kind of show the structure that could exist between categories. OK. And that's what I'm trying to do in this episode is if is, is if you want to formalize intuition, have something to anchor on a good story that has a level of precision to it, then we need something that can show the connection between categories.
Because again, I believe intuition is essentially analogy making. Analogy making is really finding connections between categories as I've explained. And there's a way to talk about the connection between categories precisely using meta mathematics using category theory. That's the take home message is that there is a way that we can come up with a more precise story of what intuition is. It's it's intuition is a structural connection between categories.
Think of it as lines connecting the categories that we create in our minds and and the categories that exist in a situation. You can actually show what those lines are with category theory, right? You can draw them out, you can depict them visually and, and they can, they can essentially open based certain meta mathematical rules, you know, whether that's compositional, other things like this.
Um But, but being able to lay down what intuition is via this categorical definition of analogy making gives us the ability to have more precise kind of definitions and, and to formalize our notion of intuition. OK. So, so being able to, so, so it doesn't have to be oh analogy kind of has this, this better connection to reality because you know, evolutionary gave it to us. And, and so it must have a better connection and just kind of leave it at that, that's all true.
But if that still kind of sounds a little hand waving and you want something more formal to anchor on, you can say, well, that connection between the categories in our mind in a situation must be connections between categories. It must be uh the these kinds of, you know, what are called isomorphic relations that exist between categories. And we can express that through category theory, we can look at what those relations are.
We can see that there is a structure to intuition, a more rigorously precisely defined structure. It's a real thing. And you, and you can see now more formally what the cost of precision would be because if you choose to be precise with symbols, this is akin to removing lines between categories, right? An intuition is like making an analogy, making analogy is like connecting lines between categories.
If you strip context, those lines are the context, those lines are the richness of the situation. You can imagine many, many, many lines, right? It's, it's this structure preserving that happens between two categories that allows us to make analogies to begin with, which allows us to have intuition. OK. And, and you must be degrading in some sense, the structure of intuition in order to create a precisely defined symbol uh uh in, in, in order to be precise about something.
And so what can we take from all this? Um Now that we've kind of delved a little bit technically there, let's let's step back and just think about what this might mean, you know, going forward in our real lives. Well, I think that the the real take home message here is that intuition must lead. OK. Now, math, I just want to be clear here, right? Math is an important aspect of how we go about explaining things, right?
Math has given us the ability to state things unequivocally uh which helps land signs on some ultimate current state that can be updated as needed. And it doesn't just have to be math, it be any, any kind of precision that we bring to something. Anytime we strip away the con text, you know, we kind of make a sterile, compartmentalized version of something we do it so that it's harder to be misinterpreted. There is a real value to that.
And, and and you know, when it comes to mathematics, I would argue perhaps the greatest value is that you can actually encode those, those equations into computer code. And then you can see how, you know these, these kind of precise mechanisms play out as we run things towards infinity. Right? There's this whole window of reality that gets opened before us when we, when we kind of translate mathematics into computation.
So, so math is, is hugely important has done, you know, great things for us throughout history and and does have a good level of rigor to it. I'm not saying math is not rigorous, but the true rigor I would argue is not in mathematics, you know, or facts or logic or any type of precision because it's not, you know, using math as an example, it's not the math that we're after. It's our connection to reality.
That's really what it means to be rigorous about something you should be rigorously as close as possible to what is actually happening. And I argue that it's intuition that has the best connection by far to reality. So intuition really has to lead. OK. What we gain from precision we lose in context. OK. Now, of course, we know all models are wrong as the saying goes, but it's not simply because they are imperfect reflections of reality.
Our models are precise arguments made with symbols and labels and those constructs are by definition structurally sparse, right? Precision will always be inadequate when it comes to capturing what's really happening. This means that we must let intuition lead an inability to articulate what we feel is not an excuse to bypass emotions. For the sake of avoiding misinterpretation. Our intuitions will lead us to profound insights, insights that we can indeed anchor in beautiful mathematics.
As long as we keep in mind that it's not the math that truly speaks to reality. It's something far more rigorous, the original intuitions that started it all. When we look upon a mathematical description of something, we are looking at how someone chose to anchor their intuitions, we are not looking at some quote unquote language of nature as though low dimensional symbolism speaks to nature's true machinery. Mathematics is a story one that grasps nature at a glancing angle.
I think the current paradigm of science and engineering forgets this and and sees mathematics as the ultimate end goal. Math is not the end goal. It never can be. It is our intuitions that hold the truth of any situation. It is our intuitions that we are chasing math must serve those not the other way around. We should never let conscious reasoning supersede our natural ability to connect to what is real. It is the connection that matters.
It is that fantastically intricate internal structure that somehow remains untouched between ideas and observations intuition must always have the last say that's it for this episode. If you'd like to take a slightly deeper dive on this topic. I write more technical versions of this material on both medium and substack. You can find them at medium dot com slash nontrivial and Sean mcclure dot substack dot com. So go ahead and check those out as always. Thanks for listening until next time.