Teaching The Logic Of Scientific Discovery With Games – Part 1: Feynman Chess

Posted on 2012/02/29

0


Feature Image - Teaching The Logic Of Scientific Discovery With Games - Part 1 Feynman Chess

“It’s a great huge game of chess that’s being played – all over the world – if this is the world at all, you know.”

Alice, in Lewis Carroll’s ‘Through the Looking-Glass’[1]

Introduction

There are many different analogies, models, and games used by teachers to try to explain what scientists do. This can be valuable, but we must be careful about what these analogies are suggesting. For example, I don’t think it’s very useful to offer a model that presents a natural history of science, or describe what scientists’ think they are doing.

Many scientists throughout history have made claims beyond their abilities, supposing to possess certain knowledge about the world, only to be shown incorrect by later experiments and observations. So, we might be cautious to take present-day scientists on their word when they reveal what they think they are able to do!

Rather, a goal of much greater value aims at imparting a philosophy of science that describes the relationship between humans and the world: presenting the logical limits of what is possible to know, what it means for us to have ‘scientific knowledge‘, and how we can acquire it.

In a series of articles, I will present an analysis of a number of these models, viewed from the critical rationalist perspective of scientific inquiry.

Feynman Chess

One ways that’s kind of a fun analogy to try to get some idea of what we’re doing in trying to understand nature, (this complicated array of moving things which constitutes “the world”) is to imagine that the ‘gods’ are playing some great game like chess …

We do not know what the rules of the game are (the rules of the pieces moving); all we are allowed to do is to watch the playing. Of course, if we watch long enough, we may eventually catch on to a few of the rules. The rules of the game are what we mean by fundamental physics.

So, you might discover, after a bit for example, that when there’s only one bishop around on the board that the bishop maintains its colour. Later on you might discover that the law of the bishop is that it moves on the diagonal, which would explain the law that you understood before (that it maintains it’s colour). And that would be analogous to when we discover one law and the later find a deeper understanding of it.

So, without being able to follow the details, we can always check our idea about the bishop’s motion by finding out whether it is always on a red square. Of course it will be, for a long time, until all of a sudden we find that it is on a black square (what happened of course, is that in the meantime it was captured, another pawn crossed for queening, and it turned into a bishop on a black square). That is the way it is in physics. For a long time we will have a rule that works excellently in an over-all way, even when we cannot follow the details, and then some time we may discover a new rule.

We’re always…(in the fundamental physics)… trying to investigate those things in which we don’t understand the conclusions. We’re not trying to check all the time our conclusions – after we’ve checked them enough, we’re okay. The thing that doesn’t fit is the thing that’s the most interesting. The part that doesn’t go to what you expected.[2] [3]

Richard Feynman

Advanced Apologies

Serious admirers of Richard Feynman might be horrified to hear that the above passage has been edited from two separate sources: a lecture to undergraduates at Caltec from 1961, and a BBC ‘Horizon’ interview from 1981.

Who am I to edit the work on one of the greatest science educators of our age, you may well ask? I hope, as you read on, you will see it less as a Frankenstein act, collecting body parts from different owners; and more of a chop-shop operation, assembling the best parts of several cars from the same factory.

This is still Feynman; just speaking across the decades. For those who wish to read him in original, I have included the full texts in an appendix below. 

I first encountered Feynman’s analogy of scientific inquiry (which I shall refer to as ‘Feynman Chess’) when I bought the audio recordings of his Caltec lectures. It immediately struck me as a deeply imaginative and charasmatic model of what scientists are up to.

After teaching with it for a few years, I came to be more hesitant. This was partly due to an increased understanding of the philosophical works of Karl Popper, and partly the result of dwelling on the assumptions of the analogy for several years.

Of course, any good teacher must remember that however tempting it may be to blindly accept an analogy, based on the suspicion that its creator was a genius (and who doesn’t suspect Feynman of that?!), all new ideas should be approach critically, whoever first thought of them.

Indeed, I admire Feynman for how he thought, as well as what he thought. The last thing he wrote on the blackboard in his Caltec office was, “What I cannot create, I do not understand,” which I consider to be the most admirable aphorism I have ever read.[4]

Teaching with this attitude can have great benefits. Firstly, it can help avoid student (and teacher) misconceptions incurred from a blind belief in a faulty model. More broadly, your students can learn to appreciate the value of criticizing ideas, influenced by your example.

So, that’s my advanced apology for criticizing Feynman!

Analysis of Feynman Chess

Undisputed Aspects

  • As with the human goal of scientific inquiry, Feynman Chess presents the observer with a challenge to search for patterns (or rules, theories, hypothesis, conjectures…) that describe and predict the universe.
  • In both Feynman Chess and scientific inquiry, we do not know the patterns in advance, but must guess them.
  • Feynman Chess also presents obvious examples that show the value of constructing conservation laws: that is, seeking properties of the game that do not change with time. For example, in chess, the bishops display ‘colour conservation’. Each time we look, we’ll find the same bishop on the same colour square.
  • Learning occurs only when our guessed patterns are falsified by experiment. We don’t learn anything at all when our guessed rules agree with experiment. For instance, in the bishop-capture/pawn-promotion scenario, we learn that the ‘bishop conservation’ guess is, in some way, false.

Metaphysical Considerations

Although Feynman doesn’t mention this, the analogy offers an opportunity to discuss metaphysical presumptions, and their relationship with scientific hypotheses.

For instance, you might assume, as you do with chess, that there are objects (the pieces) that move around the square on the board.

However, we can look at the whole thing from another perspective. Imagine that the squares are the only objects, and the pieces are the properties of the squares. So, to use the ‘colour conservation’ rule from before, we might reinterpret this to mean that ‘bishopness‘ is a property that is only transfered diagonally between squares!

Is this a bishop object moving from one square to another,
or is it the property ‘bishopness’, which transfers diagonally though blue squares?!

So there can be several different metaphysical interpretations of the same guessed pattern. What matters is that the prediction fits with the observation, regardless how the observations or the thoery are interpreted. If a theory is refuted, we abandon it along with our favouite interpretation.

One of the most glaring metaphysical presumption can be heard when people speak about ‘energy‘ and ‘mass‘. Mass, it seems to most, is associated with objects; while energy is a property of objects (a function of its position and its motion). However, later down the line in physics education, you might learn about Einstein’s famous equation that presents the equivalence between mass and energy. It suggests that mass and energy must either both be objects, or both be properties.

So which is it!? I’m not going to answer the question, but remember that whatever answer you arrive at, it has little to do with the success of the predictions a theory makes.

Points for Discussion

[In discussing the disputable elements of Feynman's analogy, I quoted from the three sources found in the Appendix below.]

Are there objective rules?

By the construction of the analogy, we are told that Feynman Chess has objective rules. However, there is no such hint provided by our own universe. We might suppose that the universe operates according to rules. Indeed, it might be a necessary methodological conjecture (as the goal is to construct patterns), but we can not be certain of this guess, unlike in Feynman Chess.

Passive Observation and Isolation

In the description of his analogy, Feynman writes about how one might perform an experiment:

“…[t]here may be situations where nature has arranged, or we arrange nature, to be simple and to have so few parts that we can predict exactly what will happen, and thus we can check how our rules work. (In one corner of the board there may be only a few chess pieces at work, and that we can figure out exactly.)”

However, if Feynman Chess is a model of scientific inquiry, then it is one where the inquiry is passive. We are mere observers of the game, and cannot construct experiments to test our conjectures. Thus, although Feynman is correct to point out that often ‘we arrange nature’ to test our ideas, this is impossible in his Feynman Chess.

The passive inquiry of Feynman Chess is analogous to astronomy and cosmology. Although we construct many expeirments on (or near) earth, our influence is very small, and we must observe most of the universe passively (although, of course, we do construct better observing devices). However, it is worth noticing that our observations need not be undirected; we can direct our attention, motivated by the desire to falsify a specific prediction, or make detailed studies of unexpected phenomena.

Also, Feynman suggests one key way to learn from nature is during moments when “in one corner of the board there may be only a few chess pieces at work”. I think he means that the useful experiments (created or not) are the ones where we investigate only one aspect of nature (whatever that may mean).

Students will be familiar with this idea from when they speak of designing an experiment to be a ‘fair test’.

However, in real chess, this is not necessary, because moves occur one at a time. Perhaps in Feynman Chess multiple pieces move at once, and the rules of capture involve the coordination of multiple pieces. However, Feynman doesn’t make this explicit when describing his model.

More than just rules

At several points in the telling of this analogy, Feynman speaks about the rules of the game and the decisions of the players of the game.

Similarly, in a regular game of chess, we intuitively think that there are two distinct aspects to the pieces movements:

  • The permissible (‘legal’) movements of the pieces, and the rules for capture.
  • The decisions each player makes when it is their turn, given they often have multiple permissible moves.
These ideas have analogues in Feynman Chess. For instance, he writes:

Even if we knew every rule, however, we might not be able to understand why a particular move is made in the game, merely because it is too complicated and our minds are limited. If you play chess you must know that it is easy to learn all the rules, and yet it is often very hard to select the best move or to understand why a player moves as he does. So it is in nature, only much more so; but we may be able at least to find all the rules.

And later in the same lecture:

“The third way to tell whether our ideas are right is relatively crude but probably the most powerful of them all. That is, by rough approximation. While we may not be able to tell why Alekhine moves this particular piece, perhaps we can roughly understand that he is gathering his pieces around the king to protect it, more or less, since that is the sensible thing to do in the circumstances. In the same way, we can often understand nature, more or less, without being able to see what every little piece is doing, in terms of our understanding of the game.”

These passages are very problematic, and raise questions concerning the following elements of the analogy, and their possible correspondence with scientific inquiry:

  • The agents-players (and our knowledge of them)
  • The objective of the game (and our knowledge that there is an objective)
  • The specific criteria a player must meet to satisfy this objective (and our knowledge of these criteria)

These players of the game certainly aren’t us; Feynman makes it clear from the beginning that our place in the model is as observers watching the game.

And, for actual scientific inquiry, we do not (as methodological rule) presuppose the universe it exhibits ‘intent’, or has goals. In other words, we don’t consider teleological explanations to be useful.

At the very least, we certainly don’t assume such an esoteric teleology where two demiurges engage in battle, toying with the universe in a competition of their own creation. We are in the business of looking for patterns, and avoid such paranoid speculations.

Thus, Feynman seems to offer a false analogy when he speaks of how “it is often very hard to select the best move or to understand why a player moves as he does”.

Indeed, Feynman would frequently admonish sloppy explanations from people who spoke of nature ‘trying’, ‘wanting’ or ‘liking’ to do thing. All we know (and can know) is that it just does things!

Why then would Feynman choose to include all this in his model? I think it is because he wanted to include the idea that there are some things we can’t practically (or possibly) know about the world.

If I am correct about this, then I think there are two potential interpretations of why Feynman spoke about the agency of the players of the game:

  1. The player’s decisions represent the problem we face when the initial conditions of a system are too numerous and complicated to measure or calculate. Despite these practical constraints, we can still construct pseudo-patterns that describe, approximately, the general behavior of a system at large.
  2. The decisions of the two agents represent quantum indeterminacy, and the ‘rules’ represent the rest of physics. The rules could either be: (A) the equations of quantum mechanics, or (B) the classical approximations as a consequence of quantum decoherence.

I think the second interpretation seems to be more fitting with the model, yet I suppose that Feynman actually intended the first interpretation.

The reason I am hesitant to accuse Feynman of (2) is because it would be out of character for him to include such difficult ideas in an introductory model. Indeed, for his Caltec undergraduate course, Feynman Chess appeared in the second lecture entitled “Basic Physics” – 35 lecutres prior the first mention of quantum mechanics!

Additionally, what Feynman actually said in the original lecture deviates in one key respect from the lecture notes that ended up in the book. From the audio recording he says:

Aside from not knowing all the rules, what we really can explain in terms of those rules is very, very limited… because, in almost all situations… the situation’s so enormously complicated that we cannot follow how everybody’s doing all the rules, and what’s gonna happen next.

So, he seems to be speaking quite definitively about (1) rather than (2). Nevertheless, both interpretations are problematic as they attempt to model problems of scientific-knowledge with the design of the game, rather than what the observer knows (or does not know) about the game.

Regardless of these broad criticisms, both interpretations offer interesting points of discussion, which I outline below.

(1) As mentioned, I think that Feynman intended to include some analogy of how knowing the rules of the game is not always useful for predicting real life situations.

For instance, we have a pretty good idea of the rules (mechanics) for an oxygen atom bouncing around inside a sealed contained. However, even for a small container of oxygen gas, it would be far too difficult and laborious to measure the positions of all the oxygen particles and predict their motions from these basic rules!

There are billions of billions of particles and billions of collisions every second. So neither could we measure the initial positions, masses and velocities of all these particles, nor could we compute all their collisions from our guessed rules.

However, we can make macro-observations of a large collection of particles. For instance, we can measure and predict gas pressure, without having to worry about that we know this ‘pressure’ is really a rapid successions of particle collisions. These aren’t really fundamental patterns, but approximate pseudo-patterns that are practically useful.

This is a useful idea to impart to students that are about to embark on trying to understand the world. Without it, they might get very disappointed that their education doesn’t begin with an immediate description of the latest knowledge about the fundamental laws!

However, Feynman Chess presents this insight with a false analogy. It implies that the focus of our inquiry should be directed at trying to predict the tactics of a one of the players of the game. And, although we cannot do this exactly, we might be able to make macro-observations, just like we do with gas pressure.

I think this is what he means when he says, “we can roughly understand that he is gathering his pieces around the king to protect it, more or less, since that is the sensible thing to do in the circumstances”.

Our difficulties arise from the overwhelming myriad occurrences of the supposed rules, whereas Feynman presents an element of to Feynman Chess that is entirely separate from the rules of movement or capture of the pieces.

It is inescapable that Feynman is offering an intuitively indeterminate model, because it contains agents with intuitive free-will as deciding the movement of the pieces within the universe.

(2) Quantum indeterminacy is described brilliantly by Feynman in one of his later Caltec Lectures. He writes:

We would like to emphasize a very important difference between classical and quantum mechanics.  We have been talking about the probability that an electron will arrive in a given circumstance.  We have implied that in our experimental arrangement (or even in the best possible one) it would be impossible to predict exactly what would happen.  We can only predict the odds!  This would mean, if it were true, that physics has given up on the problem of trying to predict exactly what will happen in a definite circumstance.  Yes!  Physics has given up.  We do not know how to predict what would happen in a given circumstance, and we believe now that it is impossible, that the only thing that can be predicted is the probability of different events.  It must be recognized that this is a retrenchment in our earlier ideal of understanding nature.  It may be a backward step, but no one has seen a way to avoid it.[6]

So, although we might suppose to have some success guessing the rules of our universe, we appear to be excluded from ever guessing a pattern that could successfully determine all future events. Quantum mechanics seems to indicate that our relationship with the universe makes it impossible learn total predictability.

Similarly, we might think that someone who doesn’t know the rules of regular chess might be able to successfully guess the ‘rules of the game’ (the ‘L’ shape of the knight, for instance) but they won’t be able to successfully guess a pattern that describes the arrangement of the board from one move to the next.

Why do we suppose this? It’s because we think we have a choice about which piece to move next, and this doesn’t cohere to any rule.

Thus, Feynman’s analogy raises a lot of interesting questions about our own supposed free will, and the possible ‘will of God’ that might act upon the universe. However, if there are demiurges playing a quantum game, they will be restricted by quantum probabilities. They might have choices in the game, but they are limited and boring!

The Chess Devil

Imagine an observer watching you play chess against an opponent. After watching you for a while, he thinks he has a pattern that describes all your previous moves, and can predict your future moves as well. What could happen if he tells you this pattern? Well, you would presume that you’d be able to willfully contradict it, by choosing some other move.

Our notions on free will relies on an assumption that we are always able to contradict any pattern someone (including ourselves) might construction that claims to predict our choices.

Of course, sometimes, there are no choices in chess, such as when a player is in check with only one escape, and therefore has only one permissible move. This is how very simple ‘chess puzzles’ are constructed, where you try to figure out a series of ‘checks’ leading to a ‘checkmate’ so that your opponent’s turns present only one permissible move, up until their defeat.

But most of chess is not like this. Most of the time, we think we have a choice about which move to make.

To begin unpacking this interpretation Feynman Chess, we might ask what we mean by ‘determined‘. I think we mean something like, “any two situations with identical beginnings working under identical laws will unroll in the same way.”[5]

We suppose that regular chess isn’t determined because one arrangement of pieces on the board entails multiple legal ‘next moves’. Just think of the opening moves between two players in a best of 5 – they might pick different opening moves for each game.

Is there any way the observer could successfully predict all the movements of chess games between these two players? What might they need to know? We’ll let’s imagine that the person watching is also allowed to know all the occurrences within the brains of the two players leading up to their choice, in addition to the movements on the chess board.

However, the observer is still lacking some vital data. Our brains are not isolated – they are greatly influence by outside stimuli beyond the present state of the board. So, let’s go the whole distance and presume that the observer is knowledgeable of all this too.

Thus we can imagine the observer is an modified Laplacian demon. Whereas Laplace’s demon was all-seeing of one instant, and all-knowing of the patterns of the universe, our demon is all seeing for the duration of the game, but does not know the rules of the game (which has now expanded from one chess game to the whole universe!).

So, with this amount of information, is it possible for the demon to successfully predict all the movements of chess games between these two players?

Well, this is the essential problem of free will. If the world is determinate, then the demon could figure it out, with sufficient imagination. We would have to conclude we do not have free will as we might suppose we do. [Now consider the potential paradox of the demon telling you his prediction. With the new stimuli, even if the universe is determined, this might be the impetus to change your mind! Of course, the demon would know this. So can he fully predict the future or not?!]

What would it mean if the demon could never succeed in its goal? Does this mean the world in indeterminate?

Well, not necessarily. It might just be that the demon is excluded from being able to figure it out! In the same way, it would seem we are excluded from knowing everything about our universe. However, this does not entail that if we ‘ran the tape back’ and started the universe from the same initial conditions, we should expect a different outcome. The problem is not what the universe is like, but what we are able to know.

Conclusions

In this analysis, I have attempted to stretch Feynman Chess a little further than perhaps Feynman had intended. Obviously, I have wandered through far too many ideas for an introductory model.

Yet, it is always wise for a teacher to have stretched a model beyond its intended scope, before presenting it to students.

I hope that even if you have disagreed with my assessments of Feynman Chess, you will nevertheless appreciate the value of the philosophy of science, and teaching the logic of scientific discovery.

References

[1] Carroll, Lewis (1871)Through the Looking-Glass And What Alice Found (drawings by John Tenniel)

[2] Feynman, R P & Leighton, R B & Sands, M (1964) ‘Feynman Lectures on Physics’ Volume 1 Chapter 2: Basic Physics (Section 2-2)

[3] Feynman, Richard (1981) ‘The Pleasure of Finding Things Out’ Interview for BBC Horizon/PBS Nova

[4] Hawking, Stephen (2009) ‘The Universe in a Nutshell’ New York: Bantam Spectra

[5] Holton, Richard (2012) From Determinism to Fatalism, and How to Stop It [forthcoming in Andy Clark, Julian Kiverstein and Tillman Vierkant (eds.) 'Decomposing the Will' (Oxford University Press)]

[6] Feynman, R P & Leighton, R B & Sands, M (1964) ‘Feynman Lectures on Physics’ Volume 1 Chapter 37: Quantum Behavior p. 10

Appendix

[Extract from: Feynman, R P & Leighton, R B & Sands, M (1964) ‘Feynman Lectures on Physics’, Volume 1 Chapter 2: Basic Physics (Section 2-2) Addison-Wesley]

[The hear the original lecture of September 29th 1961, click on the link below. Note that Feynman does not say exactly the same thing as printed in the book]

‘What do we mean by “understanding” something? We can imagine that this complicated array of moving things which constitutes “the world” is something like a great chess game being played by the gods, and we are observers of the game. We do not know what the rules of the game are; all we are allowed to do is to watch the playing. Of course, if we watch long enough, we may eventually catch on to a few of the rules. The rules of the game are what we mean by fundamental physics. Even if we knew every rule, however, we might not be able to understand why a particular move is made in the game, merely because it is too complicated and our minds are limited. If you play chess you must know that it is easy to learn all the rules, and yet it is often very hard to select the best move or to understand why a player moves as he does. So it is in nature, only much more so; but we may be able at least to find all the rules. Actually, we do not have all the rules now. (Every once in a while something like castling is going on that we still do not understand.) Aside from not knowing all of the rules, what we really can explain in terms of those rules is very limited, because almost all situations are so enormously complicated that we cannot follow the plays of the game using the rules, much less tell what is going to happen next. We must, therefore, limit ourselves to the more basic question of the rules of the game. If we know the rules, we consider that we “understand” the world.

How can we tell whether the rules which we “guess” at are really right if we cannot analyze the game very well? There are, roughly speaking, three ways. First, there may be situations where nature has arranged, or we arrange nature, to be simple and to have so few parts that we can predict exactly what will happen, and thus we can check how our rules work. (In one corner of the board there may be only a few chess pieces at work, and that we can figure out exactly.) A second good way to check rules is in terms of less specific rules derived from them. For example, the rule on the move of a bishop on a chessboard is that it moves only on the diagonal. One can deduce, no matter how many moves may be made, that a certain bishop will always be on a red square. So, without being able to follow the details, we can always check our idea about the bishop’s motion by finding out whether it is always on a red square. Of course it will be, for a long time, until all of a sudden we find that it is on a black square (what happened of course, is that in the meantime it was captured, another pawn crossed for queening, and it turned into a bishop on a black square). That is the way it is in physics. For a long time we will have a rule that works excellently in an over-all way, even when we cannot follow the details, and then some time we may discover a new rule. From the point of view of basic physics, the most interesting phenomena are of course in the new places, the places where the rules do not work—not the places where they do work! That is the way in which we discover new rules.

The third way to tell whether our ideas are right is relatively crude but probably the most powerful of them all. That is, by rough approximation. While we may not be able to tell why Alekhine moves this particular piece, perhaps we can roughly understand that he is gathering his pieces around the king to protect it, more or less, since that is the sensible thing to do in the circumstances. In the same way, we can often understand nature, more or less, without being able to see what every little piece is doing, in terms of our understanding of the game.’

———————–

Transcript from: Feynman, Richard (1973) ‘Take the World from Another Point of View’ PBS: NOVA Interview

[The watch a video of Feynman saying this, click on the video below - quote begins @ 11:27]

“The world is strange. The whole universe is very strange. But, you see, when you look at the details, and you find out that the rules are very simple – of the game – the mechanical rules by which you can figure out exactly what’s going to happen when the situation’s simple.

It’s again this chess game business. If you were in just a corner where only a few pieces are involve, you can work out exactly what should happen. And you can always do that when there’s only a few pieces involved. So you know you understand it. And yet, in the real game, it’s so many pieces, you can’t figure out what’s going to happen…”

———————–

[Transcript from: Feynman, Richard (1981) ‘The Pleasure of Finding Things Out’ Interview for BBC Horizon/PBS Nova]

[The see a video of Feynman saying this, click on the video below - quote begins @ 27:18]

“One ways that’s kind of a fun analogy to try to get some idea of what we’re doing in trying to understand nature, is to imagine that the ‘gods’ are playing some great game like chess – let’s say a chess game – and you don’t know the rules of the game but you are allowed to look at the board, at least from time to time, and in a little corner perhaps. And from these observations, you try to figure out what the rules are of the game, what the rules of the pieces moving.

So, you might discover, after a bit for example, that when there’s only one bishop around on the board that the bishop maintains its colour. Later on you might discover that the law of the bishop is that it moves on the diagonal, which would explain the law that you understood before (that it maintains it’s colour). And that would be analogous to when we discover one law and the later find a deeper understanding of it.

Then things can happen… everything’s going good. You got all the laws; it looks very good. And then all of a sudden, some strange new phenomenon occurs in some corner. So, you begin to investigate that – to look for it. It’s castling! Something that you didn’t expect.

We’re always, by the way (in the fundamental physics), always trying to investigate those things in which we don’t understand the conclusions. We’re not trying to check all the time our conclusions – after we’ve checked them enough, we’re okay. The thing that doesn’t fit is the thing that’s the most interesting. The part that doesn’t go to what you expected.

Also, we could have revolutions in physics. After you’ve noticed that the bishops maintain their colour, and they go along the diagonals, and so on, for such a long time (and everybody knows that that’s true) then you suddenly discover one day, in some chess game, that the bishop doesn’t maintain its colour. It changes it colour.

Only later do you discover the new possibility. That the bishop is captured and that a pawn went all the way down to the queen’s end to produce a new bishop. That could happen, but you didn’t know it.

And so it’s very analogous to the way our laws are. They sometimes look positive; they keep on working. Then all of a sudden, some little gimmick shows that they’re wrong, and then we have to investigate the conditions under which this bishop change of colour happened, and so forth. And gradually learn the new rule that explains it more deeply.

Unlike the chess game though… in the case of the chess game, the rules become more complicated as you go along. But in the physics, when you discover new things, it looks more simple. It appears on the whole to be more complicated because we learn about a greater experience. That is, we learn about more particles and new things. And so the laws look complicated again.

But if you realize all the time, what’s kind of wonderful, is as we expand our experience into wilder and wilder regions of experience every once and while we have these integrations in which everything is pulled together in a unification which it turns out to simpler than it looked before.”