Computational Conversations EP#7 | The power of abstraction through history

Hello, I'm Conrad Wolfram.

And I'm John McLuhan.

And this week's computational conversation is on the power of abstraction through history. We want to talk through, in all walks of life, some of the things that we take for granted that might not be thought of as abstract or that we might not realize that we use every day, and then talk about how computation fits in with this, including it being one of those great abstractions. So I think it's sort of like where to kick off this. I mean, I guess let's just start with one kind of abstraction, which I think is. I always find fascinating, is money. And money, you know, we just use every day. We take it for granted. But presumably, originally, things were basically bartered for each other, you know, and the idea of something that abstracted that out. And there are levels of this abstraction, right? There's the concept of a value that is not, you know, in itself an object, so to speak. But of course, originally it was to some extent an object because coins actually were valued to the silver or gold in the coin.

And then that's just the levels of abstraction that money's gone through, that it's become progressively more abstracted. But I think the concept you picked there is a good one because it's doing exactly what abstraction needs to do, which is it's throwing away detail that is unimportant. So if you are bartering, whether it's a pig or a cow doesn't really matter. It's the intrinsic value. So it's getting rid of unnecessary detail, and it's allowing you to do things that were hard to do before. The abstraction that allows you to decide is one pig worth three cows or one cow or 10 chickens, you know, that value balancing. So I think when, if you look at other kinds of abstraction, they're all about throwing away the detail you don't care about in order to be able to work more effectively with the kind of intrinsic nature that you're trying to capture by the abstraction. I don't know if you think that's a fair summary of.

Yeah, I think it's funny what intrinsic nature is or isn't. I mean, I suppose one of the things that's exciting is that abstraction generates a standardization of sorts. So you've got a cow and you've got a pig, and you're trying to barter them for each other. It's like, cows are worth this, and pigs are worth this, and we still have that concept. What would they go at auction? But the idea that you can talk about both of them in the same nomenclature with the same way of representing them. This idea of standardization seems a key thing running through abstraction, I think.

Yeah, I think we've. One of the basic goals is an abstraction that gives you. Well, I'm going to fall straight to computation in this case, because this is a numerical thing, but things that you can do with the abstraction that. That work across all the domains that you can apply the abstraction to. So the idea that I can owe money. Well, that applies regardless of what I owe money for. And it's. Yeah.

As opposed to owing cows, for example.

Yes, yes. Wouldn't that be a terrible world if you had to pay up in the thing that you initially.

I also just want to go through. Just think through for a moment these layers, because I think this is also common of different abstractions. We should talk about some others soon to. To see whether it fits. But you can imagine, I don't know the full history of money. But to start with, it was like, okay, can we have a value of metal, gold, silver, whatever it is that represents the value of or that is valued at the same as the cow or the set of them? And of course, then they're numerically added together. So you can have multiple things added, as you would have done in bartering, but it was sort of more difficult because the objects were too big, each of them. So obviously you had the idea of the material was worth that. Then I think that changed into this sort of concept that actually you could have coins that were materially, you know, worth less. But in a sense, there's some entity that is representing the value. So here the bank of England.

But all that is, in a way, is you're inventing abstractions upon the abstraction. So once you've got a language to talk about something like value, because you've got invented money now, you can start abstracting on the concept of value, and you can start having things like debt or options or all these other things.

That, you know, you jumped several. Several steps ahead.

Right.

Because after coins, there are then notes. I mean, literary pieces of paper. And I think on the British one, actually, I should pull one out. But I think it says, I promise to pay the bearer of this note the following. So the note is a reference for.

Five pounds of silver?

Yes, originally, presumably. And somebody who's much more trained in this will know the details of this. But now, of course, we've gone further to cryptocurrencies where there isn't even a physical representation of them at all.

I don't think you see this process through all abstractions. If you take one of the most basic ones that, you know, we invented words, words are an abstraction. That the fact that you have a name so I don't have to constantly describe you when I'm talking about something, I can use the name. And we all have that as an interchange to say, when I say, Conrad, we know who we're talking about. We've then built abstractions on abstractions. So now we have words to talk about words. We have things like adjective and verb, which are absolutely a direct abstraction on top of the abstraction. And then we start talking about things like abstract grammars that don't even care about.

You're even zooming ahead. I mean, I'm fascinated by the idea of the Alphabet. Right. I mean, it's a pretty big leap, if you think about it, to go from hieroglyphs, which, you know, were sort of a sort of abstracted picture of what things look like so that you could all agree that this meant a bird, to at least the, you know, various Western and other cases where the idea of taking the word and not making a huge melee of characters out of that word, you know, sort of abstracted hieroglyphs, but actually turning that into a small number of elements that together could be put together to make any number of different words, that seems a huge leap.

Well, actually, I'm going to slightly disagree and say that hieroglyphics are a great example of somehow sometimes these things are just incremental, that there was a time where you told a story by literally doing the story in pictures and you would just have the story retold like a cartoon. And actually, hieroglyphs are a transition because I believe. I'm not a linguist, but my belief is that hieroglyphs sometimes literally represent the thing you see, but very often they represent sounds that are the sound of the word of the thing you see. So if you have a picture of a bird, it would be kind of like a bur sound or whatever it is in ancient Egyptian. And so it's almost. It's almost that halfway between sometimes representing the thing and sometimes representing a sound. That is the stepping stone to our more modern alphabets where we literally just try and capture the sounds and the letters don't mean anything in themselves. But what I'd like to do is to take this as an opportunity to jump into good and bad abstractions, because I think alphabets are a good example of that. But we have historical baggage with those. And there are some ways in which being speakers of British English, where they're very bad abstractions on the sounds because we have all kinds of bizarre spellings that you just have to know. And they're failing in their job of abstracting. They're becoming at a word level, an abstraction. But the letters have sometimes lost their abstraction of the sounds they were supposed to represent and they've become kind of muddled. And if you go back to our normal world of computation, I think there's all kinds of examples where you can see good and bad vers of abstraction. Like Roman numerals were numbers, but they're not very useful abstraction because they don't lend themselves to things like arithmetic in columns. And so there are much better number systems and worse number systems. And if you get more complex, you've got things like Leibniz notation in calculus. The DY dx, which is sometimes for certain operations, is quite useful to remind yourself how you do things. But I'm sure you went through the same mistakes of doing things where you treated them like a fraction back when you were first learning calculus and your teacher wagged his finger at you and said, no, no, they're not actually fractions, they just look like fractions.

Actually, I think Richard Feynman used to want to cancel the Ds. He used to say in DY.

Yes, exactly. So that's as a place where the metaphor of the fraction is breaking down and actually becomes dangerous and misleading.

Well, that's the notation of the fraction, because I think again, you're putting together different levels of abstraction. There's the abstraction of the idea of a fraction, then there's the actual notation. And I mean, back to the Alphabet thing a bit. I mean, I think. And the pronunciation. It's interesting, actually. Think about this. I hadn't thought of this before, but there's the notation and then there's a sort of another layer that's been put on. So sometimes there are sort of parallel layers of abstraction operating together, sometimes they're hierarchical layers. So, for example, if you take there's letters, there's words, there's sentences, there's paragraphs, there's chapters, there's books, different levels, all in different cases. Now we're all using letters to make words and sentences often, although even that's reduced in text messages and things. Whether you write old book with chapters is again, not always there, but in parallel to that. There's how you speak the abstraction, how you write the abstraction, which is sort of parallel rather than hierarchical. And it's. I mean, the word thing is Interesting, because of course, we have. If you look at Asian languages, they don't abstract for the most part to, you know, to letters. They're still with characters. And that has plus and minuses. And I think one of the things I'm also interested in is how the way in which you abstract changes your pattern thinking. So there's a funny possible feedback loop here of, you know, kind of like the way that you choose to abstract is somewhat driven by the way you think about things. And then the way the abstraction is compared to what you're actually trying to do drives your thinking somewhat. And I wonder whether one can see this in human languages. And I think you can. I mean, it's often said that. I think we probably did this in the previous podcast in English, for example. I think we covered this before, but as always said, the practical words tend to be Germanic and the emotional words tend to be French. And they have a slightly different style in how they are. And I think when you look at how people from those different areas have big generalization, there's some slightly different thinking patterns, and who knows how those interplay.

Yeah. So I think one of the things you have to think about is what is the uses of the abstraction. And, you know, you can have abstractions that are entirely personal. You know, when I write code, I have a picture in my head that is an abstraction of the structure of the code that I've got. And that's just me assembling information in a way that I can process. So it's purely for its internal computational benefit. But a lot of abstractions are about having a common language to be able to interchange information. And there it starts to really matter how you notate or verbalize the abstraction that, you know, I might want to explain to you the structure of some narrative text, but we have to have in common language like metaphor, so that I'm able to talk about it. And there, if we don't have well structured ways of representing the abstraction well, they're intrinsically linked. I can't talk about an abstraction unless I have words for it or notation. And the words relate to each other in a way that is somehow consistent and usable, or it's just confusing. I mean, here's a very modern converse of that, which is when you look at machine learning, where the very nature of what's going on inside neural networks is they're constructing an internal abstraction that they see a lot of data and they start to build structures that say, oh, I often see the things that look like this and things that look like that. And they relate in certain ways that I'm starting to see patterns to now when we do that as humans, we have been taught various abstractions. That interchange has already happened and we've been programmed to look for certain abstractions. If I'm saying it's a dog or a cat, I will use language you would understand to explain why when I say, well, it's got really pointy ears, so that's more cat like. And its nose is very pointy, that's more cat like. And know what other things, maybe it's got sharper claws, you know, these are the things I'm pointing out to you are abstractions on this whole mess of visual imagery that I'm seeing when I look at a cat. If you look at the insides of neural networks, it's like you've just met somebody from some cut off civilization in a jungle who not only speaks a different language, but thinks completely differently. And when they try and tell you that they've just seen a tiger and how they know it's a tiger, they're using words. Not only do you not know the word, but you don't even have a matching word. It's not even like a notation thing. You don't have the concept for the thing that they saw that told them it was a tiger. And that's what goes on all the time in machine learning. You get these features.

But I think there's another point you bring up which is the explicitness of the abstraction to at least you, the human.

And you mean to what extent is this a human?

Well, I mean, is it visible? I mean, it might be there's some abstraction happening in a machine and you as the human user of the machine don't really know it, but it's happening anyway. Or it might be that you directly as a human are interacting with the abstraction. And sometimes you're doing it very naturally, like with words and letters and so forth, because you've been trained for a long time to do that. Sometimes it's hidden from you. I mean, a lot of machinery we use, there's, you know, I press the accelerator in my car and all sorts of things are happening, being calculated, all sorts of absolute. I mean, the idea that the accelerator actually an interesting question. How is the accelerator linked to how fast I'm driving? Actually quite complicated, right? It's not a very obvious abstraction. Now that was initially driven by what could be done because, I mean, it was physically connected to, you know, the fuel system. Now of course, it could be completely differently, graded and effectively, some of what's done now is to backport to what you're used to. So I mean, one of the.

I think my point is that it's not just about the visibility of stuff. Stuff can be a mystery, but if it was revealed to you, go, okay, I get that that's electronics, I studied those. So I understand the abstraction of electronics that's going on in between. Even when you expose what's going on inside a neural network and say, tada, this is the feature it's looking for, we simply aren't trained to recognize that as a feature because it's not in our training, in our lexicon, we have.

No word and we have different machinery. Right? I mean, so in the same way as I was just saying that, you know, the way you think, in my view, is partly what causes the kind of abstraction and partly the abstraction guides the way you think. I think that very central is the actual machinery for the abstraction. When the machinery is, you know, human brains, that's a different bunch of stuff to play with than when it's, you know, modern computers. And so what's good, what works well and fast and so forth in those is different. And so that affects the abstraction. And one of the things with language, and I want to come on to computer language as well as human language, but one of the things I find interesting is the extent to which your prior knowledge, your way of thinking, the culture you're in, affects the kind of abstraction, affects what's possible, affects how well, you know, how do I choose to abstract the meaning I'm trying to get across to you now into a sentence that's pre program. I mean, that's a whole set of layers of things. Now why am I doing that? Well, there are a bunch of constraints I have. I want to do it reasonably efficiently. I hope what I'm saying is clear to everybody listening to it, that it's convincing and interesting, et cetera, et cetera. So those are all constraints for me. I have a certain brain power to be able to do that and a certain formulation of ability to do that. I'm not very poetic. So a poet might produce more nicely sounding words, et cetera, et cetera. I don't know. A computer might look at it a rather different way. Now we're trying to make computers behave a bit more human, like in a sense, by introducing more abstraction layers. And I think we need to do that for a moment. Why does anyone care about this abstraction? We talked about it a bit at the beginning. I think it's so that one can make more progress in decisions, in ideas, in all areas. I mean, we were talking about money at the beginning. You can grow, trade a lot if you have abstract money. Now, of course, you also get new problems. And there are all sorts of things where people have money that doesn't really exist. I mean, as in it's not tethered to anything. And so you end up with huge crashes and things like that, which weren't just a figment of abstract money, the tulip bubble and so forth. But they are very related to more abstraction means there's sort of more layers before you see something's gone wrong.

But I think when you get into this kind of layer on abstractions, you've got both a powerful benefit and a dangerous problem. As with most of the tools, which. Yes. Which is that you are able to think at much higher and higher levels without getting bogged down the detail. It's just like a lot of maths. It's about getting rid of complexity to make the problem small enough to be able to think about in our relatively small brains. And so by the time you're somebody in a bank working on concepts like collateralized debt obligations, there's no way you could talk about trading those kinds of objects. If you were down at the details of counting little piles of gold coins and putting them in different piles that represent different people's interests. You have to talk about things at a much higher level. But equally, that isolation from what in the end is going on behind the scenes and forgetting the limitations of the abstraction is a great danger. And where things go wrong, even very simple things, I think that we take something like one plus one equals two, and we're so used to numbers and integers that we don't stop to think about the limitations of that abstraction. But if you stop and think, there's lots of places where one plus one doesn't equal two. If I put two clouds together, I might just end up with a cloud because they merge. They don't have a notion of unitariness that 2 fraction of integers relies upon. Or digging a hole in a hole.

I've just got a bigger hole, by the way. Let's just talk explicitly about some of the abstractions within maths. Obviously maths itself is a whole abstraction.

But then you get abstractions upon abstractions just right.

So zero elsewhere. Zero and infinity are both fascinating abstractions. And infinity, as you say, has many interesting. Infinity plus infinity is always a great thing to play with young children when they understand, sort of roughly the concept of infinity. What's infinity divided by infinity? And et cetera, different orders, different directions, et cetera. I mean, to name a few others, calculus. I mean the idea of abstracting down to ever smaller things to add them.

Up so that until you abstract away the finiteness of them.

Yes.

What I think is fascinating, one of the great things about abstraction, when it works well, is you work it out in one field. And when you have some completely different situation, think, what if I apply that abstraction and that kind of thinking to this? It works. And so one for me that is kind of fascinating is probability. That the abstraction of probability can be applied to almost opposite problems that you have. The what will happen in the future? Which is a probability question. But it's also. You can apply the same maths to what happened in the past. Given my limited knowledge, can I best figure out what happened? And when you stop and think about it, they're just opposite things. And yet the same abstraction tends to work in both cases. You just have to be very careful about what you mean as you go into the step of abstracting it and de. Abstracting it. As you know, I did this chick with my daughters when they were little, when they understood probability. And I hold out a coin and say, what's the chance it's going to be heads? And they say a half. And I flip it and put it on my hand and say, what's the probability now? And they'd say a half. And I'd look at it and say, what's the probability now? And they're starting to dawn on them that each of these is a different question. What will happen? What did happen? I know what happened. And I'm asking you what you know about what happened. And finally I would say, it's heads. What's the probability now? And at that point my daughter switched from saying a half to saying, oh, you always tell the truth. So it's one which I thought was.

Very nice trusting Dorothy than I have.

But the same question four times has very four different meanings because it's the root of abstracting into the probability is different indeed.

And I think what's. I mean, this idea that it carries over to so many different areas and newer and newer areas you find new areas if it's a successful abstraction. I think what started off as sort of numerical maths in some cases ended up into algebra, ended up into calculus and so forth, because somehow successful abstractions go way beyond. And then at some point they become natural and we all forget they were abstractions to start with. Money is a great Example, by the way, another example I thought was interesting is countries. The idea of countries and what entity size you associate with as a person, what community size do you associate with? It's actually quite funny. I remember asking somebody who had been. I won't say, I think it was a prime minister of one of the European countries who's very pro European Union. And this was before the Scottish referendum. And I said to him, what entity size do you think people like to associate with? Because it's not. I mean, countries in the sort of modern formulation of countries are pretty new in the size that they are. And, you know, at the time we. We were having lots of people wanting to bring, you know, Scotland and Basque region of Spain, so forth, you know, all trying to think of breaking off and so forth. And he was saying, no, no, I mean, you know, Europe is the entity size. I said, well, you know, from a sort of counting point of view, it seems to make sense. But is that how people think about it? So. So abstraction can have a sort of. Sometimes there's a sort of maybe a apparent irrationality about how one wants to abstract because it feels better one way or another, one associates with it. And of course, you know, there are more tensions now about these very big blocks. And that I think is something that, you know, again, wasn't fully taken into account because the abstraction of, you know, I'm part of this country doesn't necessarily fit everybody's viewpoint.

Yeah, Well, I think as we head intowards politics, politics is one of the places that I think abstraction is somehow sometimes least useful. And maybe this also touches on countries that we use things like political parties as a way of shortcutting the complexity of our world to say, separate the right answer from the wrong answer. But the idea that if you look at the package of things that go with the right or the left, why those things are grouped together and why large numbers of people agree all of those things are right and all of the other sides are wrong because what were they done? Party is a way of shortcutting the thinking.

Correct.

And arguably you could say the same thing about countries that a lot of, you know, of at least the last thousand years wars have been driven by identifying with the country over the other, thus, and them, and you're either with us or against. This kind of thinking is a shortcut. That's unhelpful.

Well, I mean, there's a certain sort of tribalism, right, which has been intrinsic.

Which is an abstraction. You're one of us or you're one of Them.

That's right. And I mean, there's a certain sort of early form of abstraction in that to some extent that there's a sort of leadership structure, which you can argue. And of course, companies and organizations are a form of abstraction too, and they have different organizational systems, different basis sets, as you call it. For maths. One thing I want to touch on before we round out is whether explicit abstraction is critical in maths and computations. We go forward. So this has been a very current topic because of LLMs and AI. So there are some people who've been saying, now we can just speak to the computer in English or whatever language. Somehow we don't need abstract code explicitly. I mean, maybe the computer is writing something, or as you say, it's got its neural networks, which are of course, abstractions of different sorts. But we as humans don't need to abstract. We don't need to work with an abstract basis. And whether this is really the case, I'm very, very dismissive of this point of view because I think that abstraction brings with it. Explicit abstraction brings with it several things. It brings a clarity of thought sometimes and the clarity of description. It can be harder to make because you've got to kind of learn the process of abstraction. So in terms of computer languages, you've got to learn a language to abstract two and how that works. And that's got a bunch of baggage with it. So I think there are many cases where you may not do the whole process. You may get an LLM or AI to help you write that. But I think the idea that you never need explicit abstraction, somehow in that context or in other cases where we got used to abstraction, seems quite false to me in understanding how progress has been made in human history.

In the end, it comes back to, I think, what are the benefits of abstraction? And whenever we're thinking about things like how we work with a computer, where it has to be explicit because computers are dumb, you know, we always want to focus on the things that it gives us. Can I express at a higher level, freed from detail to what I want to achieve, what I know, what I don't know, all of those things. So that requires a certain amount of clarity. Is it something that. Is it a abstraction that I can have in common with the one I'm conversing with? So whether that's the computer and LLMs help here, so do programming languages or the other humans who have to see my work? Does it unlock some processes, thinking, computation, algorithm that applied in a different situation but can apply in My situation. And I think those are the three goals you always want to think about with abstraction. Is it clear, is it simple and is it useful?

Yeah. Do you need it explicitly? I mean, I do think that machinery can help make abstract, I suppose. Okay, let me put another way. I think that. Is this true or not? I think that abstraction, the idea of building abstractions and using them has been up until now, quintessentially human capability. And I use the word quintessentially quite carefully. Quintessentially doesn't mean it feels like it's imbued in being a human. Doesn't necessarily mean, you know, animals don't do any abstraction or whatever. But so now I think we. One of the. I suppose the leaps forward that we might see is that computers, machines are getting much better at doing this abstraction for us in a way that wasn't really working before.

Well, actually, I think all of science is teaching us to be a bit more humble about these things we thought were intrinsically human, that machine learning has shown us computers generating their own abstractions. All the studies of animals that we try and find something that only we do tool use. Well, there's crows that use tools. There's a study just a few weeks ago that claimed that African elephants call each other by name. So there they've come up with the abstraction that I don't just mean elephant or you over there. I have. I know which elephant. And I've got a shared token that I can exchange with others to, you.

Know, I mean, I love the level.

Of abstraction, but it's still there.

Well, I mean, there's an even nicer one. There was. I can't remember what they were, but some sort of either monkeys or the like, who. There was a place tourists visited and they'd worked out that women tend to have longer hair and they were pulling it out of the tourists and they were using it to floss their teeth.

Which I think is really advanced abstraction of a tool that is good for teeth that is suddenly recontextualized. Away from twigs to hair is a form of abstraction thinking, isn't it? I don't know if they think about it that way or it just feels nice in their hands and.

Well, I mean, obviously they thought it was. Had a utility. So I think abstraction to me is something that is quite useful to think about explicitly, both in what we're setting up and why we're using it, because I think it also tells us something about where the failure modes are and where the power really is. I Mean, I think, for example, if you think about that with money, our starting point actually, if there was a little bit more thought generally every day about. I'm sure in the hallowed areas of central banks and things, they do think about this, and particularly with modern new techniques like cryptocurrencies and things. But I think sometimes it gets lost in everyday banking type thing and we get things like the banking crash, which I think where people had sort of, I don't know, they'd forgotten about the abstraction versus the reality in some sense. I think that was one of the drive and there were all sorts of drivers and problems. But. So I think one danger with abstraction is that you forget that it is an abstraction for the reality and that you kind of forget about the reality that you're in. So it's got a power because it is away from the reality. It means you can take lots of realities and represent with an abstraction, but it's a danger when you forget that there was a reality it was supposed to be associated with.

And money is a fabulous example of that. Is there going to be a day where everyone with Bitcoin suddenly wakes up and says, actually they're just numbers, it's not worth anything as long as we all kind of agree the abstraction, then the abstraction gains power.

And there has to be some basis. It's the gold standard. Whether it's how difficult it is to mine a bitcoin and how much computing power it takes. That's kind of there has to be some value that it makes it hard to do in that sort of case, I think with mathematics.

Well, that's kind of like back to the mechanics of can you do something useful with the abstraction like transfer value? If you can't compute or at least intellectually process the things in a way that is useful, then they're not useful. So if there wasn't the mechanics of, of mining and transfer of Bitcoin, it would still be bitcoin, it would still be an abstraction, but it wouldn't be a useful one anymore.

I mean, if maths were easy, right, as an abstraction, I don't think that would be negative. I mean, except if you're using the.

Difficulty is the tool in this case, it's perversely the wrong way around. The fact that the maths is hard in the case of Bitcoin is the useful tool because it checks the uniqueness.

In that particular case. But maths in general, not associated with Bitcoin or money. I think that you don't need an int, it doesn't need to be intrinsically difficult to be useful. It may happen to be, but. Well, we should probably round that. I mean, I think. What's our conclusion from this? I think that abstraction is one of the most powerful drivers in human history, has many.

And that's useful since we've been abstracting maths and computation for 30 years. So it's nice to be reassured there's some value in it.

Well, indeed. And I think maths and computation is one of the. Is right up there as one of the. I mean, and perhaps we're only a little way into it as we are at the moment, but I think, you know, in. In 2024, I think it may end up being one of the world's most empowering abstractions ever. I mean, you know, along with. Along with some of the ones we've been talking about. But it's already very powerful. Even the fact that we've built machines of computers that can do anything effectively in terms of the universal computing machines, that in itself is in a sense powerful to do with the abstraction that maths and computation brought to the world. And I think we're also believing that explicit abstraction looked at by humans will continue to be important. Though there may be many cases where in the past you explicitly had to deal with the abstraction, then in the future you don't. But that it isn't going away. The idea of abstract, the idea of humans handling it directly isn't something that's going away anytime soon.

Yes, and I think we stay ahead of the animals by simply getting more and more layers of abstraction and stay ahead of them and potentially the computers as well.

An abstraction tower.

Yes.

Well, I hope that's been an interesting discussion and look forward to seeing you next time on Computational Conversations.

Goodbye, everyone.