Bad Epistemology

By Mark Goldblatt

“There is nothing new under the sun,” the author of Ecclesiastes assured us more than two millennia ago. Among the questions he glosses over, however, is how the sun got there in the first place. Which brings us to the Big Bang. In case you haven’t noticed, the origin of the world has been much in the news recently despite the fact, or perhaps because of the fact, that it’s the oldest story in the Book.

It’s scientists who’ve been digging into creation of late, and doing so very publicly and very provocatively. This isn’t necessarily a good thing, either for science or for the rest of us, because the topic is an epistemological minefield and scientists are notoriously bad epistemologists, and, as a general rule, people who go digging in minefields tend to produce less than optimal outcomes.

The latest scientist to take on creation in a loud way is Neil deGrasse Tyson, host of the hit TV show Cosmos. In an interview with the Huffington Post, Tyson had this to say about the universe’s beginning:

“If you take the universe all the way back to the Big Bang, well, the entire universe was really small. So now you take the shotgun wedding – quantum physics and general relativity. In that shotgun wedding, if you follow through with all the predictions quantum physics gives you, it allows multiple bubbles to form – one of which is our universe. These are sorts of fluctuations in the quantum foam. Quantum physics fluctuates all the time. But now the fluctuations are not just particles coming into and out of existence, which happens all the time. It’s whole universes coming into and out of existence.”

You hardly know where to begin in addressing the errors in this passage. But let’s start with the idea of the “universe.” By definition, the universe is the totality of what was, what is, and what can be. It is, as the philosophers say, co-extensive with being. Whatever is possible, whatever is not a contradiction in terms, belongs to the universe. You and I are part of the universe. George Washington is part of the universe, despite the fact that he’s no longer with us. Even unicorns are part of the universe – although, as far as we know, they don’t actually exist. Real beings. Imaginary beings. Present beings. Past beings. If it’s an actual or potential being, if you can sensibly conjugate the verb “to be” in its vicinity, it’s part of the universe. Why? Because the whole purpose of a concept like “universe” is to encompass everything under one umbrella term—i.e., every thing.

You don’t have to fill up chalkboards with arcane mathematical formulae to know that there are no other universes; you just have to observe the prefix uni and to understand language. If there are other dimensions, or other modes of existence, past, present or future, they are – again, by definition – part of this universe, our universe. They are part of the world in its entirety, which is another way of saying they are part of the totality of what was, what is, and what can be.

But let’s be charitable to Tyson, who’s a scientist and not a philosopher, and who may not be acquainted with the ancient philosophical tradition of defining the universe as co-extensive with being. Let’s assume that when he talks about “whole universes coming into and out of existence,” all he means are those “bubbles” of matter and energy which lend themselves occasionally to detection and measurement – one of which we happen to inhabit (and which, mirabile dictu, is the only bubble we can currently detect and measure). Tyson insists that the existence of our world can be accounted for by the “shotgun wedding” of “quantum physics and general relativity.” How? By a “fluctuation in the quantum foam.”

Well, no.

If “quantum foam” exists, then matter and energy already exist. Hence, the universe already exists – even in the limited sense of the word “universe” that Tyson (again, to be charitable) intends. The question then becomes: How did that quantum foam come into existence? I don’t know how Tyson would answer that. I suppose he could say, Well, it was just always there. Except if he did, he’d be digging himself an even deeper philosophical hole. Because “always,” means “without a beginning” – which is another way of saying that an infinite amount of time has passed during which the quantum foam has existed.

You don’t want to go down that road, however, because you wind up knee deep in the problem of actual infinity. To be infinite, by definition, is to be unfinishable. Lots of things are potentially infinite – a line, for example, can potentially be extended infinitely in both directions. But you can never reach infinity. You can never make the potential actual. You never quite get to a point, as you’re extending the line, when you can kick back, have a beer, and say, “Okay, now at last the line is infinite.” You can’t finish what is, by definition, unfinishable.

Now think about time as a line going backwards. To say that an infinite amount of time has passed during which quantum foam has existed is to say that we’ve gotten to that point of infinity. Here we are, and past time is now infinite. We’ve finished the unfinishable. That’s what logicians call a contradiction in terms. If you’re forced to suppose the existence of a contradiction in terms – if, in other words, you’re forced to suppose that a thing is what it cannot be – you know you’ve gone wrong. It’s back to the drawing board.

(Around now is when some guy in the balcony who took differential calculus as an undergrad will usually shout down that infinity is used routinely by mathematicians…so why can’t something be actually infinite? It’s true that infinity is a handy mathematical concept, but it is used as an ideal limit, a value that can be approached but never reached. Sound familiar? Calculus Guy will usually be followed to the mike by Set Theory Guy, who’ll assert that the German mathematician Georg Cantor “proved” that sets with an infinite number of members exist. There are many flaws with Cantor’s work – too many to detail here – but suffice it to say that Cantor never spoke of an infinite set consisting of real objects like marbles or melons or moon orbits; rather, his infinite sets were always populated by abstractions – like the set of natural numbers, or the set of all sets. Indeed, Cantor himself acknowledged the distinction between, on the one hand, “reality” and “quantity,” and, on the other hand, “number” and “set.”)

But there’s an even more serious problem with Tyson’s statement: He’s implicitly rejecting the law of causality. He insists that whole universes – perhaps an infinite number of whole universes? – come into and out of existence all the time due to fluctuations in the quantum foam. But what causes the fluctuations? Tyson implies nothing does; the process just “happens all the time.” Here, he’s echoing Stephen Hawking, the world’s most famous living scientist, who in 2012 stated, “Spontaneous creation is the reason there is something rather than nothing, why the Universe exists, why we exist.” Weaving together the latest insights from string theory, gravitational theory and quantum theory, Hawking concluded, “The universe can and will create itself from nothing.”

Well, no.

If there’s nothing – to note only the most howling flaw in Hawking’s logic – if there’s literally no thing, then there’s no universe to create itself. (Hawking’s “nothing” is a more brutally illogical formulation than Tyson’s pre-existing “quantum foam.”) But even allowing that a miniscule speck of something, or some thing, ends up big-banging into the universe, you still have to account for that change from speck to world. A speck is what it is, and it will remain what it is, absent a causal force acting upon it; so says the law of causality. (The law of causality can be thought of as the law of conservation of identity.) Blaming the Big Bang on a quantum fluctuation gets you nowhere because you still need a cause for the fluctuation. The mathematical quirks of theoretical physics do not constitute exceptions to the laws of thought.

There are no exceptions, none whatsoever, to the laws of thought, including the law of causality. Nothing – literally, no thing – is exempted. No matter how distant. No matter how exotic. No matter how teensy-weensy. Not even the alleged topsy-turviness of quantum physics lets you wriggle free of them, despite what many contemporary scientists seem to think.

As I mentioned earlier, scientists, even brilliant ones like Tyson and Hawking, are often bad epistemologists. They tend to climb out on limbs and want to saw off the branches they’re sitting on. There is indeed a raging debate among scientists about whether causality holds in the quantum realm. So let me settle it for them: Yes, it holds. I say this with total confidence because their debate isn’t really about causality but about predictability – whether a complete mathematical description, if it were possible, of the conditions of a subatomic particle would allow an observer to predict the future state of that particle. The answer may well be no; there may be no possible mathematics to make such a prediction. But the fact that you’ve hit a mathematical dead end doesn’t mean that the change occurs uncaused; there is still some causal force behind it.

Mathematics describes forces but is not itself a force. If I drop a baseball down an elevator shaft, mathematical formulae can tell me where it will be and predict, with great accuracy, how it will behave at a given moment; but mathematical formulae are not causing the baseball to descend. Gravity is doing that. We may not know, and may never know, the exact nature of the force that governs the behavior of subatomic particles – call it the Quantum Fairy if you like – but there is a force operating on those particles, causing those probability equations to work out. There’s a force beyond the law of averages – which, again, is a mathematical description, not a causal agent – ensuring that order emerges out of apparent randomness. You can take that to the bank.

The irony is that scientists are more than happy to reason this way in other contexts. For example, they speculate that “dark matter” must exist to account for the state of the visible universe, and then go looking for it, because they know there must be a cause for every observable effect. Yet utter the word “quantum” and all bets are off. Ancient priests ridiculed the quest to find causal connections, invoking divine will as the solution to all mysteries – the so-called “God of the Gaps”; modern scientists echo them now with the “Quantum of the Gaps.” Can’t come up with a cause for subatomic behavior? No worries, that’s just quantum strangeness for you!

Except causality is science. It is the sine qua non of science. Having shimmied out on the limb of theoretical physics, scientists want to saw off the branch of causality that got them there. But the quantum realm isn’t a logic-free zone. “One cannot get around the assumption of reality,” Einstein wrote to Erwin Schrodinger in 1950, and reality is “independent of what is experimentally established.” Subatomic particles may be paradoxical little devils, fiendishly hard to pin down, but nothing about them rewrites the laws of thought. They are what they are, and they stay what they are, unless they are caused to change. If your equations force you to abandon your axioms, you’re screwed. It’s time to climb back down the tree and start over.

Which returns us, at last, to the reason we’re having this conversation in the first place: the Big Bang theory. On the one hand, scientists like Tyson and Hawking love the Big Bang theory because it provides an elegantly straightforward explanation for the existence of the universe, and because it seems borne out by more and more sophisticated experimental data. On the other hand, scientists like Tyson and Hawking hate the Big Bang theory because it points squarely to the first line of Genesis: “In the beginning, God created the heavens and the earth.” So desperate are they to avoid the slightest taint of religious sentiment that they’re willing to embrace literal absurdities – an actual infinitude, or an uncaused effect – to avoid the slippery slope back to the First Cause or Unmoved Mover favored by medieval theologians.

My message to Tyson and Hawking, therefore, is one of comity rather than enmity: C’mon in, boys. The water’s fine.

 

Mark Goldblatt teaches religious history at Fashion Institute of Technology of the State University of New York. His two most recent books are “The Unrequited”, a literary mystery, and “Twerp”, a novel for young readers.

 

 

  • http://www.patheos.com/blogs/daylightatheism Adam Lee

    If an actual infinity can’t exist in reality, then doesn’t this also prove that God is not omniscient, because that would require knowledge of an actually infinite set of true propositions?

  • Nathaniel

    So lets see, on one end I got a Christian apologist with a focus on philosophy. On the other, we have a trained astrophysicist. And now I need to judge who is more credible on the topic of astrophysics.

    Decisions decisions.

  • Dorfl

    The ‘bad epistemology’ that you complain about basically boils down to one realisation on the part of scientists: That if you say “a priori, the universe has to work like this”, the universe is not going to say “oh I’m sorry, I didn’t realise” and change the way it works to accommodate you. Your use of the phrase ‘laws of thought’ is new, but otherwise the content of this post is very, very familiar.

    But let’s be charitable to Tyson, who’s a scientist and not a philosopher, and who may not be acquainted with the ancient philosophical tradition of defining the universe as co-extensive with being.

    Professionals in different fields will define words in different ways. There is no more charity in assuming that a scientist uses the scientist’s rather than the philosopher’s definition of ‘universe’ than there is in assuming a programmer and a carpenter mean different things by ‘file’.

    You don’t want to go down that road, however, because you wind up knee deep in the problem of actual infinity. To be infinite, by definition, is to be unfinishable. Lots of things are potentially infinite – a line, for example, can potentially be extended infinitely in both directions.

    From your comments about Calculus Guy and Set Theory Guy, it’s clear that people have explained to you repeatedly why you’re wrong about this, but you went “Nuh-uh” and ignored them, so I’m writing this more for the sake of other readers than in any real hope of changing your mind:

    Right between 8 am and 9 am there is 8.30 am. Right between 8.30 and 8 there is 8.15, while right between 8.30 and 9 there is 8.45. Between all these is 8.07.30, 8.22.30, 8.37.30 and 8.52.30. Obviously I can continue this forever, pointing out more and more points in time between 8 am and 9 am. That is, there is not a finite number of those points. Equally obviously, there is nothing potential about those moments. They are not being added by my process of pointing them out. They are all actually there. And you will pass through them soon, or already have, depending what time zone you’re in.

    In short, actual infinities exist and we encounter them all the time. They only seem tricky for the same reason that maths students find it so difficult to accept that 8.99… = 9.00… – human intuition tends to make the mistake of thinking about infinity as a process, and then becoming confused about how that process can ever finish.

    If there’s nothing – to note only the most howling flaw in Hawking’s logic – if there’s literally no thing, then there’s no universe to create itself. (Hawking’s “nothing” is a more brutally illogical formulation than Tyson’s pre-existing “quantum foam.”)

    Mathematical models are the best tools we have for describing how reality works at the most fundamental level. They are certainly a much more accurate tool than ordinary speech, which was never designed for that sort of thing. So given a verbal description of a physical process (e.g. “the heat from the candle caused the ice to melt”) and a mathematical description of the same process (e.g. the equations describing the time evolution of the candle-ice system and their solutions), I will pretty much always assume that the mathematical description is closer to the underlying reality. People who aren’t good at mathematics tend not to like that, but there’s nothing I can do about it.

    So when you hear people like Hawking and Tyson say these kinds of things, what you’re listening to is a translation, from a language they are fluent in, to one you can understand. You may believe that you’re pointing our glaring logical contradictions when you complain about ‘howling flaws’ and ‘brutal illogic’, but you’re not. You’re just pointing out things that did not fare well in translation. If you don’t want to have to deal with translation problems, it’s up to you to learn the original language.

    A speck is what it is, and it will remain what it is, absent a causal force acting upon it; so says the law of causality.

    When you have a system which is not at maximum entropy, it’s virtually guaranteed that along at least one direction of the time axis, the entropy will increase. When this happens, it becomes possible to talk about ’causes’ and ‘effects’ instead of just ‘this thing that happened’ and ‘that thing that happened’. Otherwise, there is no causality, just things happening.

    Since it so happens that we do live in a world which is not at maximum entropy, we can pretty much always think in terms of causality. That can trick us into thinking that causality is somehow a law underlying physics, rather than an emergent property of physics.

    The mathematical quirks of theoretical physics do not constitute exceptions to the laws of thought.

    There are no exceptions, none whatsoever, to the laws of thought, including the law of causality. Nothing – literally, no thing – is exempted.

    And here we reach the big problem:

    The human brain comes with a number of built in intuitions for how the world can be expected to work. Those have been built in to us by evolution, since they allow us to form models of surroundings that are accurate enough to be very useful. Because they’ve been built by evolution, they cannot be expected to work outside of situations humans have tended to have to deal with.

    There are two ways of dealing with this. The approach settled on by Tyson, Hawking and others is to accept that outside of human-sized scales – whether of distance, time, energy, speed or something else – our intuitions are not going to be of much use, and mathematical models are likely the best guide we’ll ever get to what the underlying reality is actually like.

    The second way is to declare that those intuitions are ‘common sense’, ‘a priori knowledge’, ‘properly basic beliefs’ or – as they’re apparently calling it these days – ‘laws of thought’. This is basically a dead end, since of course the universe doesn’t care about any of that. It gives you no real option beyond stomping your feet angrily when actual things stay infinite and things keep happening without cause.

  • Donalbain

    We may not know, and may never know, the exact nature of the force that governs the behavior of subatomic particles – call it the Quantum Fairy if you like – but there is a force operating on those particles, causing those probability equations to work out.

    If you have managed to find the elusive hidden variables of quantum mechanics, please publish your work, and accept your Nobel Prize. If, on the other hand, you are just saying that you REALLY REALLY REALLY want there to be such a force, then have fun with that.

    • Michael Snow

      “We may not know, and may never know…”

      From that, how do you manage to finagle “If you have managed to find…”?

      • Donalbain

        I managed it by reading more than you did, If you continue reading, the author states, as a matter of fact, “there is a force operating on those particles, causing those probability equations to work out.” . Now, the options are that he has found evidence for such a force, in which case he will win a Nobel Prize with a certainty approaching 1, or he is just expressing a faith or a desire that there SHOULD be such a force, in which case I find that amusing.

  • Mark Goldblatt

    Dear Dorfi—

    Thank you for your thoughtful (in the literal sense, setting aside the occasional lapses into sarcasm) response. Let me deal with your objections one by one:

    1) Your variant use of Zeno’s Paradox to demonstrate the existence of an actual infinity between 8:00 and 8:30 merely shifts the problem to the existence of actual infinitesimals. I think the error comes through more obviously in geometry, but it’s equally applicable to time. By definition, any line segment contains an infinite number of points. Except there is no such thing as an actual infinitesimal. Think of it this way: the difference between an infinitesimal and zero would have to be infinitely small. So in order to actualize an infinitesimal, you’d also need to actualize the reciprocal of infinity. You would need to divide your way down, let’s say, to one “infinitieth” of an inch. But just as you can’t build your way up to infinity by addition, so too you can’t whittle your way down to infinity by division. No matter how many divisions you make, you’re no closer to an infinitieth of an inch than the moment you started whittling. As Aristotle points out, “The fact that the process of dividing never comes to an end ensures that this activity exists potentially, but not that the infinite exists separately” (Metaphysics 9:6). To return to your 8:00 to 8:30 example, there are potentially an infinite number of infinitesimally small moments, but there is no single moment that is infinitesimally small—hence, there is no actual infinity.

    2) You assert that mathematical models are better descriptions of reality than ordinary speech. But both ordinary (rational) speech and mathematical models rest on the laws of thought—which are more fundamental than either of them. If, for example, you say that two plus two equals four, and I say that two plus two does not equal four, we know that we cannot both be right. (That’s the law of non-contradiction.) We know that one of us must be right—either two plus two equals or does not equal four. (That’s the law of excluded middle.) We know that two equals two, and that four equals four. (That’s the law of identity.)

    The law of causality is an extension of the law of identity: A thing is what it is, and it will remain what it is, absent a causal force acting on it. You write, “When you have a system which is not at maximum entropy, it’s virtually guaranteed that along at least one direction of the time axis, the entropy will increase.” To which the obvious question is, why is that guaranteed? What cause drives that movement towards equilibrium? To say that it “just happens spontaneously” begs the question—it assumes you can have an effect without a cause. Of course, begging the question is a logical problem, and you seem to be asserting that mathematics is prior to logic since you write: “That can trick us into thinking that causality is somehow a law underlying physics, rather than an emergent property of physics.” Frankly, I’m not even sure what you mean by “physics” in this context. Do you mean a system of equations that describe the state of the world? Because a description does not determine the characteristics of the thing it describes; at best, it reflects those characteristics. Or by “physics” do you mean the state of the world itself: i.e., causality is an emergent property of the state of the world itself? But in that case, what are your grounds for thinking there are exceptions to it? How do you reason your way to that conclusion without invoking the laws of thought? Why is the idea of an uncaused effect less absurd than the idea of a Quantum Fairy?

    It is certainly the case that we could reason deductively and arrive at abstract, formal knowledge using only the first three laws of thought (identity, non-contradiction and excluded middle) without the law of causality: All men are mortal; Socrates is a man; therefore, Socrates is mortal. If the first two statements are true, the third must be true. Based on the first three laws of thought, we would also know—given the definitions of “two,” “four,” “plus” and “equals”—that two plus two equals four. But in order for inductive reasoning (empirical science) to work—for it to provide an accurate description of reality—we must take the law of causality as axiomatic. If we assume that identities can morph uncaused, that a thing can change into something else without a force acting on it, then we cannot describe a thing as what it is since it is constantly changing into something slightly different. A statement like “The moon orbits the earth” becomes meaningless because “the moon” is no longer one thing; it has changed (albeit in minuscule ways) in the time it takes to utter the statement. The law of causality accounts for the fact of that the moon remains the moon, despite the reality of its changing, because it supposes that every change is the result of a cause rather than an unaccountable shift in identity. Insofar as physics describes real objects—i.e., objects that exist independently of our thinking about them—it is irrevocably bound by the law of causality.

    3) Your closing paragraph does summarize the core disagreement between us. You assert that the human brain comes with built in “intuitions” about how the world works; these intuitions are a byproduct of evolution since they are useful for survival, but they cannot be expected to work “outside of situations humans have tended to have to deal with”—that is, on scales beyond our usual ken in terms of “distance, time, energy, speed or something else.” In those situations, “mathematical models are likely the best guide we’ll ever get to what the underlying reality is actually like.”

    What you’re calling “intuitions,” however, are actually the laws of thought. And what I’m asserting, yet again, is that those laws of thought underlie the “mathematical models” in which you have so much faith. Hence my point about climbing out on a limb and sawing off the branch you’re sitting on. You don’t get to those mathematical models if you don’t presuppose that a thing is what it is (the law of identity), that a thing cannot be and not be simultaneously (the law of non-contradiction), that a thing must either be or not be (the law of excluded middle), and that every effect has a cause (the law of causality). These are your axioms, the foundations not only of common sense and ordinary language but of mathematics and physics. They are the basis of rational thought and meaningful utterances in any context—regardless of the scale.

    • Dorfl

      I must say I’m enjoying this discussion a lot. I’m learning a lot from having to express my ideas explicitly – much more than I would have if we didn’t disagree on everything.

      Zeno’s paradox

      Zeno was concerned with intervals, which you are correct have to have some finite length. But I’m just talking about moments – that is, individual points in time. And the length of a point isn’t infinitesimal, it’s just an ordinary boring zero – just like the number three is not an infinitesimal interval on the real line.

      Laws of thought

      I would say that the laws of thought are laws of, well, thought: They are not rules that external reality can be required to follow. They are guidelines for how our thinking about external reality* should work.

      To give an example, a Patheos neighbour – Andrew Murtagh – made an argument a while ago about the law of non-contradiction, where he said that it’s inconceivable that you could be inside your car and not inside your car at the same time. I didn’t comment at the time, but I could very easily conceive of someone being in a state that couldn’t really be described well as either ‘inside the car’ or ‘not inside the car’, for example by hanging out of the window, or standing up in a convertible. That doesn’t make me throw away non-contradiction though, it makes me say “Huh, I guess I shouldn’t have modelled inside-car-ness as a boolean”. That is, non-contradiction doesn’t constrain how you can actually be located with relation to a car. My attempts to keep my thinking consistent with non-contradiction constrain the way I have to think about being inside cars.

      An analogy that might make this clearer is if you think about a future physicist named Kim, whose current mathematical knowledge consists of addition, and whose experimental apparatus consists of lumps of clay:

      By thinking long and hard about the matter, Kim has concluded that 1+1=2 is true and 1+1=1 is false. Doing experimental work she then discovers that if she takes one lump of clay, another lump of clay and combines them, she still only ends up with a single lump of clay – albeit a bigger one. Her response is not to conclude that her mathematical work is incorrect, and actually 1+1=1. It is to conclude that addition was not a good model for the process of combining clay lumps. After considerable experimental work, she realises that while addition was a bad model for how lump number behaves under combination, it seems to work quite well for lump weight.

      The important thing to understand here is that Kim did not demand that the lump behaviour should correspond to the rules of arithmetic, and she did not throw away her rules of arithmetic when she first failed to find a correspondence. She set up a formal system of arithmetic, demanding only that it should obey its own internal laws. She then observed external reality, trying to find some way that it seemed to map onto her formal system. It turned out that it did, but she could never have required it to.

      Science without causality

      If I understand you right, you accept that mathematics on its own works without causality. Given that, it should be easy to show how we can have physics without introducing a concept of causality. For the sake of the argument, we will take an impossibly idealised picture of how scientific progress actually works. I could make the same argument for a more realistic picture – I would just need to be even more verbose.

      Physicists put together a formal mathematical model, involving Lagrangians, differential equations, symmetries and a whole lot of other things. The model does not explicitly contain anything called causality, but it has the interesting property that something like causality arises in a number of special cases.

      They then hypothesise that external reality maps onto this model – that whatever actually is out there behaves according to the same patterns as are built into the model. This allows them to make a large number of predictions of what should be observed if the hypothesis is true, but would not necessarily be if the hypothesis were false. By Bayes’ theorem such predictions that pan out then increase the probability that the hypothesis is correct, and the predictions that don’t lower it. How much the probability is changed is something that reasonable people can disagree on, but at some point to number of successes is so high and the number of failures so low that the larger part of the scientific community agrees that the hypothesis is probably correct.

      Note that at no point did the scientists need to introduce causality. They do not claim that their model is the cause of anything, it’s just a formal mathematical structure. They made no a priori claims about external reality except that they did not know it to be impossible for it to follow the same patterns as were built into the model. They did not say anything about the cause of the predicted observations, just that they were more probable if the hypothesis were correct than if it were not.

      If somebody then asks why their model does not seem to contain causality when cause and effect manifestly exists, they can say “If you look at this special case something like what you intuitively mean by cause and effect falls out. This special case happens to correspond to the part of reality we actually live in. So causality is not something put into the model, it’s a prediction made by the model”. If that person then insists that they cannot imagine a situation where causality does not apply, they can go on to say “You really should talk to a biologist about this, but basically: your brain has evolved to have some built in software for situations that you’re likely to actually encounter. Unlike literal computer software, brain software cannot easily be switched off, so your causality-detection modules are going to keep looking for causes and effects, throwing out a constant stream of false alerts in a situation where causality doesn’t apply”.

      ‘Approximative’ is not meaningless

      In the previous discussion, I talked about a model such that every part of external reality maps onto some part of the model. This would essentially be a theory of everything, and obviously we do not currently have one**. So far we only have models that are mapped onto by some subset of reality. That is: partial, approximative models.

      This is a bit annoying, but in practise slightly fuzzy models of some small part of reality are very useful. In fact, even if we had a theory of everything we would probably stick to approximative models in most practical situations. So the knowledge that a theory of everything would not have anything labelled ‘the moon’ in it – although there would presumably by a very sharp peak of excitations of certain quantum fields in a region of spacetime – is not really a problem for me. In practise, using the phrase ‘the moon’ as a short hand for that region is very useful, and other people will tend to understand what is meant by it.

      ETA: Apparently I forgot to add my footnotes.

      * Or just thinking, if you like pure mathematics.

      ** We may find one in the future, or we may not. Some think it’s very likely we’ll get one eventually. Some reject it as obviously impossible. Given everything I’ve already said about making a priori demands on the universe, I try not to form any opinion on the subject.

    • jonathan_s_NYC

      “The law of causality is an extension of the law of identity: A thing is what it is, and it will remain what it is, absent a causal force acting on it. You write, “When you have a system which is not at maximum entropy, it’s virtually guaranteed that along at least one direction of the time axis, the entropy will increase.” To which the obvious question is, why is that guaranteed? What cause drives that movement towards equilibrium? To say that it “just happens spontaneously” begs the question—it assumes you can have an effect without a cause. ”

      I don’t have nearly the physics background of Dorfl, but I think this gets to the heart of the misunderstanding. The notion of that “a thhing is what it is, and it will remain what it is, absent a causal force acting on it” goes against the overwhelming weight of evidence that we have from physics about how the universe really works. In fact, all things that are not at a temperature of absolute zero are in a state of constant change. Change is the norm. The second law of thermodynamics can be explained by this state of change. Entropy is the tendency of the amount of energy in a system to become evenly distributed. The reason why that law is true is not because of some force – but by statistical probability – there are more high entropy states than low entropy states, so random movement tends to cause increases in entropy. An example is this: if you build a sandcastle in a box, then shake the box. The more you shake the box, the more the sand will tend to become an evenly distributed pile. Why? Because there are more possible configurations of the sand that tend to look evenly distributed than that tend to look like sandcastles, so if you just keep shaking, the random movement creates the “even pile” state.

  • Guest

    Dear Dorfi—

    Thank you for your thoughtful (in the literal sense, setting aside the occasional lapses into sarcasm) response. Let me deal with your objections one by one:

    1) Your variant use of Zeno’s Paradox to demonstrate the existence of an actual infinity between 8:00 and 8:30 merely shifts the problem to the existence of actual infinitesimals. I think the error comes through more obviously in geometry, but it’s equally applicable to time. By definition, any line segment contains an infinite number of points. Except there is no such thing as an actual infinitesimal. Think of it this way: the difference between an infinitesimal and zero would have to be infinitely small. So in order to actualize an infinitesimal, you’d also need to actualize the reciprocal of infinity. You would need to divide your way down, let’s say, to one “infinitieth” of an inch. But just as you can’t build your way up to infinity by addition, so too you can’t whittle your way down to infinity by division. No matter how many divisions you make, you’re no closer to an infinitieth of an inch than the moment you started whittling. As Aristotle points out, “The fact that the process of dividing never comes to an end ensures that this activity exists potentially, but not that the infinite exists separately” (Metaphysics 9:6). To return to your 8:00 to 8:30 example, there are potentially an infinite number of infinitesimally small moments, but there is no single moment that is infinitesimally small—hence, there is no actual infinity.

    2) You assert that mathematical models are better descriptions of reality than ordinary speech. But both ordinary (rational) speech and mathematical models rest on the laws of thought—which are more fundamental than either of them. If, for example, you say that two plus two equals four, and I say that two plus two does not equal four, we know that we cannot both be right. (That’s the law of non-contradiction.) We know that one of us must be right—either two plus two equals or does not equal four. (That’s the law of excluded middle.) We know that two equals two, and that four equals four. (That’s the law of identity.)

    The law of causality is an extension of the law of identity: A thing is what it is, and it will remain what it is, absent a causal force acting on it. You write, “When you have a system which is not at maximum entropy, it’s virtually guaranteed that along at least one direction of the time axis, the entropy will increase.” To which the obvious question is, why is that guaranteed? What cause drives that movement towards equilibrium? To say that it “just happens spontaneously” begs the question—it assumes you can have an effect without a cause. Of course, begging the question is a logical problem, and you seem to be asserting that mathematics is prior to logic since you write: “That can trick us into thinking that causality is somehow a law underlying physics, rather than an emergent property of physics.” Frankly, I’m not even sure what you mean by “physics” in this context. Do you mean a system of equations that describe the state of the world? Because a description does not determine the characteristics of the thing it describes; at best, it reflects those characteristics. Or by “physics” do you mean the state of the world itself: i.e., causality is an emergent property of the state of the world itself? But in that case, what are your grounds for thinking there are exceptions to it? How do you reason your way to that conclusion without invoking the laws of thought? Why is the idea of an uncaused effect less absurd than the idea of a Quantum Fairy?

    It is certainly the case that we could reason deductively and arrive at abstract, formal knowledge using only the first three laws of thought (identity, non-contradiction and excluded middle) without the law of causality: All men are mortal; Socrates is a man; therefore, Socrates is mortal. If the first two statements are true, the third must be true. Based on the first three laws of thought, we would also know—given the definitions of “two,” “four,” “plus” and “equals”—that two plus two equals four. But in order for inductive reasoning (empirical science) to work—for it to provide an accurate description of reality—we must take the law of causality as axiomatic. If we assume that identities can morph uncaused, that a thing can change into something else without a force acting on it, then we cannot describe a thing as what it is since it is constantly changing into something slightly different. A statement like “The moon orbits the earth” becomes meaningless because “the moon” is no longer one thing; it has changed (albeit in minuscule ways) in the time it takes to utter the statement. The law of causality accounts for the fact of that the moon remains the moon, despite the reality of its changing, because it supposes that every change is the result of a cause rather than an unaccountable shift in identity. Insofar as physics describes real objects—i.e., objects that exist independently of our thinking about them—it is irrevocably bound by the law of causality.

    3) Your closing paragraph does summarize the core disagreement between us. You assert that the human brain comes with built in “intuitions” about how the world works; these intuitions are a byproduct of evolution since they are useful for survival, but they cannot be expected to work “outside of situations humans have tended to have to deal with”—that is, on scales beyond our usual ken in terms of “distance, time, energy, speed or something else.” In those situations, “mathematical models are likely the best guide we’ll ever get to what the underlying reality is actually like.”

    What you’re calling “intuitions,” however, are actually the laws of thought. And what I’m asserting, yet again, is that those laws of thought underlie the “mathematical models” in which you have so much faith. Hence my point about climbing out on a limb and sawing off the branch you’re sitting on. You don’t get to those mathematical models if you don’t presuppose that a thing is what it is (the law of identity), that a thing cannot be and not be simultaneously (the law of non-contradiction), that a thing must either be or not be (the law of excluded middle), and that every effect has a cause (the law of causality). These are your axioms, the foundations not only of common sense and ordinary language but of mathematics and physics. They are the basis of rational thought and meaningful utterances in any context—regardless of the scale.

  • Mark Goldblatt

    I’m enjoying the conversation too (friend me on Facebook if you’re so inclined)—though I’m out of my depth as far as the physics is concerned. Indeed, if our disagreement were about physics, I’d put up a white flag. But our disagreement is not about physics; it’s about epistemology, and whether the physics of the really really small or the really really big permit exceptions to the laws of thought.

    About Zeno’s Paradox:
    If you define a moment as a point of time in which zero time passes, then time as
    defined as a sequence of moments would cease to exist. Adding zeroes, one following another, does not move you off the first zero.

    About the Laws of Thought:
    The example of the person who is both in and not-in the car underscores my point. Once you define what constitutes the state of being “in the car,”—feet
    on the floor? butt on the seat?—the contradiction disappears. But as soon as definitions have been agreed upon, to assert simultaneously that “Mary is in the car” and “Mary is not in the car” is only to say something and then unsay it. It is, literally, nonsense.

    The same point can be made with respect to Kim, who works with clay, and who concludes that one plus one equals one. Once you define whether the act of addition concerns weight or contiguous surface, the problem disappears. The very fact that you have to play with the definitions to arrive at the paradoxical result underscores the universality of the laws of thought.

    About Science Without Causality:
    You write: “Physicists put together a formal mathematical model, involving Lagrangians, differential equations, symmetries and a whole lot of other things. The model does not explicitly contain anything called causality, but it has the interesting property that something like causality arises in a number of special cases.”

    But nothing in mathematics can give rise to causality; mathematics is descriptive. If no human beings ever lived to discover mathematical laws and formulae, the world would still operate according to the same causal agencies. If, in your example, external reality—which, by definition, exists independently of the thought processes of a thinker—happens to map onto the model the physicists put together, well, then, good for them. Physicists are very good at modeling that sort of thing, and I admire them greatly. But to say that no causality exists in the physicists’ model, and then to jump from that observation to the conclusion that causality does not necessarily hold in reality, is a therefore-too-far. It would be like arguing that because computers can be programmed to play chess, and because they don’t care whether they win or lose, human chess players, who follow the same chess rules, and whose moves may even be predicted by the chess-playing software, must therefore not care whether they win or lose either.

    • Dorfl

      Thanks for the offer :-) The problem is just that I’m Swedish, so having me as a friend would mostly mean your daily news feed contained a dose of gibberish with umlauts.

      Where we disagree

      I would express my position like this:

      Thoughts should be constrained by the three laws of thought: identity, non-contradiction and excluded middle. If you find yourself having to think of something as both a and not-a, that’s a strong hint that your thinking has gone wrong somewhere.

      External reality is simply unconstrained by these laws of thought, not just outside of the region we have experience with, but everywhere. What this means is that when we think about external reality, the laws of thought do not allow us to make any demands on what reality can be like. They instead make demands on how we must go about modelling external reality, about what kind of models we can use and which simply aren’t well-defined. So when we find that a temperature is neither warm nor not-warm, the law of non-contradiction tells us we should change the way we think about temperature, perhaps modelling it as a real number (Celsius, Fahrenheit or Kelvin) instead of a boolean variable (Warm or not).

      The law of causality is more of a guideline than a law. It’s a heuristic that has been built into us by evolution. It’s very useful within a certain regime, but quickly becomes useless outside of it.

      So where we disagree seems to be about whether the laws of thought apply to anything other than thoughts, and about whether the ‘law of causality’ is a law of thought in the first place.

      Zeno’s paradox

      Numbers on the real line have zero width, yet they don’t seem to have any problem getting from three to four. The way we talk about time assumes that moments in time map onto the real line, so I don’t see why time should have any problem with that either.

      I think the problem with this part of the discussion is that what we’re discussing is fundamentally set-theoretical, and to avoid being Set Theory Guy, I have to refrain from bringing out the mental tools we need to discuss this in any rigorous way.

      In-car-ness as a boolean

      Say that we include Mary having her feet on the floor and her butt on the seat in our definition of ‘being in the car’. We then find that we have to include some amount of material between Mary and the car, since
      presumably we don’t want to require her to actually be naked to count as being in the car.

      Taking inspiration from Diogenes of Sinope, Mary then turns up wrapped in a thick layer of bubble wrap. Each day she makes the layer a bit thicker, until at some point we have to say that it’s not really well-defined whether she is in the car or not. We then come up with a newer, cleverer definition of ‘in the car’, which deals with that problem. But then Mary manages to find a grey area in this too, again putting herself in a state which is neither really ‘in the car’ or ‘not in the car’. We keep making up better and better definitions, but each time Mary finds a slight grey area in our definition, forcing us to make our definitions more and more ridiculously complicated.

      I’d say that the basic problem we’ve encountered in the we’ve made a model which treats ‘inside the car’ as a boolean variable, but the ‘inside the car-ness’ that actually exists in external reality doesn’t follow the pattern of a boolean. So external reality doesn’t map perfectly onto our model, and Mary is actively seeking out situations where the mapping onto is particularly bad.

      What the law of non-contradiction is doing in this case is helpfully telling us that our model is bad. We need to find some other way to think about being inside the car to capture this aspect of external reality.

      Kim’s model of physics

      Remember that Kim didn’t conclude 1+1=1. My point was precisely that she let her mathematical system be, and tried to find some way that external reality mapped onto it.

      The important thing is that Kim didn’t play with definitions to arrive at a paradoxical result. She had to play with them to get away from the initial paradoxical result. That is, keeping her thinking consistent with the laws of thought constrained how she could model external reality, even though the laws did not constrain reality itself.

      Emergence of cause-and-effect

      The concept of cause-and-effect, as we generally use it in everyday life, can be shown not to be built into our physical models, but to arise from them under certain conditions.

      At the most fundamental level, the laws of physics as we know them are invariant under time reversal*. That is, they don’t distinguish between ‘past’ and ‘future’, just ‘this direction along the t-axis’ and ‘that direction along the t-axis’. They also seem to be deterministic**, meaning that the state of a system at every point in time is determined by its state at any arbitrary point in time. This means that what events you choose to refer to as ’causes’ and which you choose to refer to as ‘effects’ is essentially arbitrary. If shown a description of a light atom bouncing off heavier atoms, whether you’d say “Atom A bounced off atom C because it had previously bounced off atom B” or “Atom A bounced off atom B because it had previously bounced off atom C” would probably be determined by whether the person making the description had preferred to define the t-axis so as to give the AB-collision or the AC-collision lower values of t. If that person refused to tell you which direction their t-axis was pointed, you would find you had absolutely no basis to prefer seeing any particular event as ’cause’ rather than ‘effect’.

      This obviously contradicts our everyday experience. We’d find it very strange if someone went around saying things like “The egg rolled off the table because it smashed against the floor” or “Everything is the way it is, because of the state of the universe at 13.15 o’ clock, March the 22nd, 1987″, even though this should be formally correct according to what we just said about the workings of the universe at the most fundamental level. The solution to this seeming contradiction is found in statistical mechanics.

      It turns out that if a system is not at maximum entropy (a statement which requires the system to be big enough for entropy to be a meaningful concept), then entropy is practically guaranteed to increase in at least one direction along the time axis. This means that the slope of the entropy function can be used to distinguish the two directions along the time axis. It also turns out that it’s computationally much easier to calculate the state of a system at higher entropy from the state at lower entropy, than it is to calculate the state of the system at lower entropy from the state at higher entropy. This means that life-forms that had evolved in a region of the universe where the entropy function has a slope, would tend to use the heuristic of thinking about things at lower entropy as ’causes’ and things at higher entropy as ‘effects’. This would be a very useful way of modelling things, even if it fails to capture the behaviour of reality at more fundamental levels.

      Redundancy of cause-and-effect

      To some extent, any statement about the properties of external reality on the basis of the mathematical model we have of external reality can be argued as going too far. Reasonable people can disagree about which parts of our model actually correspond to anything real, which parts are artefacts of the mathematical formalism and which parts of reality don’t map onto anything in the model.

      In this case though, we can say the following: There is nothing we observe that requires us to add causality to the model. For everything we actually observe that we’d be inclined to refer to as cause-and-effect, is described by the model once we take into account that we’re looking at systems of non-maximal entropy. The observation that the human mind seems to demand there to be cause-and-effect is also explained, once we take into account that evolution has given us a bunch of useful heuristics without any concern for whether those will be useful outside of the situations that we’ve tended to find ourselves in.

      We can ‘glue on’ causality to our model, insisting that some events are causes and other are effects, even though we cannot tell which ones are which. But we really have no reason at all to believe that there is anything in external reality that maps onto this feature of our model. The fact that nothing we’ve observed has needed that addition, should make it vastly more likely that there is in fact nothing external captured by it. (This is basically how I’d express Occam’s razor).

      I should add that this is not the first time this kind of thing happens. Physics has repeatedly ended up discovering that something which was thought to be axiomatically necessary, could actually be discarded without causing any problems – for example euclidean spacetime or absolute position.

      * I’m stretching the truth here a bit for pedagogical reasons. The laws
      of physics are actually CPT-invariant. Pretending that they are T-invariant allows us to have a discussion that captures every important aspect of another discussion that would have been much more verbose.

      ** I can discuss how this relates to the apparent randomness in measuring microscopic systems, but that would be an enormous sidetrack.

  • Dr Mike

    Caveat: I am but a novice when it comes to astrophysics; my expertise (if I may be so bold) lies in theology, psychology, and (to a lesser extent) philosophy. So forgive me the (possibly) ignorant question which follows.

    When it is said, “By definition, the universe is the totality of what was, what is, and what can be”, and that same universe is also said to be expanding (or contracting, if I understand dark matter at all), then into what or where does it expand? If the universe is all that is, was, or ever can be, then where does it go? Has someone posited that it encroaches on another, hypothetical universe or universes? Or is there something else in addition to “universes” that allows them to expand? And how can that be given the definition of “universe”?

    Again, pardon my ignorance. I just don’t understand what’s outside the universe if the universe is all that there is.

    Explanations are welcomed and desired. Thanks.

    • Dorfl

      I’m not sure how helpful this explanation is, but basically:

      We’ve got a mathematical description of how space behaves, that’s internally consistent and seems to be well mapped onto by observations. However, we want to transmit the basic idea of that description to people who don’t have the background knowledge necessary to follow the mathematics. If we want to translate the mathematics into some kind of intuitively graspable visual picture, about the best we can do is to say “space is expanding”.

      The problem is that the translation isn’t perfect. It leaves you with the feeling that there ought to be some external space for our space to expand into. This is basically just an unfortunate and hard to avoid side effect of the translation – much like a book translated from Russian into English might leave you feeling that the characters are all very annoyed with each other, because of how very terse everything they say is.

      Essentially everything I just said about “space is expanding” can also be applied to “space is curved”.

      • Dr Mike

        Thanks, Dorfl. While it doesn’t answer my question completely, it does shed more light on it and helps me think about it differently.

        • Dorfl

          Now that I’ve thought about it, I think I can give some sort of explanation of what we believe happens that doesn’t explicitly require mathematics:

          Imagine that we take two spaceships, and park them in space so that they have no velocity relative to each other at all. We also make sure to put them really far from anything else so no external forces are acting on them, and assume that they’re so light that any gravitational pull between them is completely negligible. We then sit and wait for a very long time.

          We will now see the distance between the ships slowly increase, even though they are not moving apart from one another. That is: they are not moving through space in different directions, instead more space is appearing between them. It probably sounds like these are two different ways of saying the same thing, but they’re not: we can distinguish them experimentally.

          The reason we refer to this as ‘expansion’ is that it’s roughly analogous to what would happen if you drew two dots on the surface of a ball, then inflated the ball to make it bigger. The dots would not move along the surface, but the distance between them would still increase. The part where the analogy breaks down is that the ball is still expanding in an external space, while we believe the space of our universe is just undergoing a change in its intrinsic properties.

      • jonathan_s_NYC

        The issue with translating physics into non-math is one I noticed and don’t see commented on enough. I do not have an advanced physics background, but I learned special relativity in AP Physics. I remember that when the teacher tried to just explain special relativity, it seemed totally nonsensical. Then when I simply followed the equations, it seemed totally sensible. It was a very strange sensation.

  • Mark Goldblatt

    Too bad about the umlauts and gibberish, but I’m nevertheless glad to have made your acquaintance.

    The core of our disagreement now seems to be whether the law of causality exists in the mind alone, or in both the mind and in reality; you are taking Hume’s position (causality is strictly a mental law), and I’m taking the anti-Hume position (causality is both a mental and an ontological law).

    As I indicated earlier, the further the conversation strays from philosophy and epistemology to physics, the more uneven the playing field between us becomes—I am compelled simply to trust you with the implications of experimental data and accustom myself to vocabulary and usages particular to an alien subject matter—but I’m game to have one more go at this.

    Your point about time reversal, and the arbitrariness of cause-effect relationships—an effect can be seen as a cause and vice versa if time is moving backwards—might alter the truth value of a statement such as “Event A is the cause of Event B,” but it would not violate the basic law that change does not occur without a cause. You write, “We’d find it very strange if someone went around saying things like ‘The egg rolled off the table because it smashed against the floor.’” That does pretty sound strange in our world. But in Backwards World, where such things happen, it wouldn’t sound strange at all because in such a world gravity would be the natural tendency of a more massive body (such as the earth) to repel a smaller body (such as the egg), causing it to fly back onto the slightly tilted tabletop and roll towards the highest point. Gravity would still be the cause; it would just work in reverse. Entropy would naturally decrease, and things would naturally organize themselves into greater and greater complexity. Fifteen rolling billiard balls would come together with sufficient force to knock the cue ball back into the pool stick, and the guy holding the stick would slowly back away from the table thinking, “Why should I play pool when my girlfriend is waiting to have sex with me?” (Joke.)

    As I noted in my original essay: “[Physicists] speculate that ‘dark matter’ must exist to account for the state of the visible universe, and then go looking for it, because they know there must be a cause for every observable effect.” If, as you say, causality is merely a law of the mind but not necessarily a law of the universe, isn’t the physicists’ search for dark matter the height of irrationality?You don’t need something like dark matter to account for the measurable state of things; things just are what they are, and they can change into other things
    without a cause, right? Human activity is causing climate change? Pah! It’s only our collective imagination that posits a necessary connection between phenomena—the burning of fossil fuels, the increase in CO2, and the warming of the planet—that are merely temporally contiguous.

    For that matter, is physics itself a description of reality, or is it merely a way of organizing our thoughts into aesthetically pleasing patterns? Do real phenomena “cause” the results of our experiments, or do our instruments respond arbitrarily, changing from one state to another state without cause, providing us with no information about a world that may or may not exist independently of our impressions of it? (After all, if the law of causality is only an “in-here” law rather than an “out-there” law, how do we know our impressions of the existence of an “out-there” aren’t just popping into our head uncaused?)

    Sure feels like solipsism to me.

    In any event, time in this vicinity keeps moving in one direction—forward—and I’m falling further and further behind in other writing projects, but I’ve learned a lot in this exchange. If you’re ever in the space-time continuum located around Manhattan, please do look me up. –MG

    • Dorfl

      Likewise, and thanks for the invitation!

      Since I just finished my thesis, and it will be a while before the next batch of phd student positions open up, I currently have unlimited time for arguing on the internet, but I realise most people don’t have that luxury. So I’ll try not to make this a long essay that demands a response, but just clarify some last things:

      The problem with talking about gravity as a cause of anything, is that the law of gravity is basically the statement “masses attract, as described by the mathematical formula so-and-so”. So saying that gravity causes masses to attract is basically just saying “masses attract, because they do”. Now, you can derive the law of gravity from the curvature of space, as described by General Relativity. I suppose you could then say that GR causes gravity if you wanted to. The problem is that you’d still be starting out from an equation that just said “the curvature of space is related to the stress-energy tensor in the following way, because it is”. Also, I think talking about laws of physics causing other laws would run into the same problems as with cause-and-effect in time: it will tend to be a matter of personal taste what you consider to be cause and what you consider to be effect*.

      Remember that while I’ve thrown out cause-and-effect as a fundamental thing, I’m still saying that external reality appears to follow patterns, and that cause-and-effect emerges when you use higher levels of description – as long as the thermodynamics of the system you’re describing is right. So I can still say “the relationship between the velocity curve of stars in a galaxy and the mass distribution violates what the pattern has previously appeared to be. Either we’re mistaken about the pattern, we’ve mismeasured the velocities, or there is a bunch of matter that we can’t see”. I can also say “at this level of description, we may say that humans are causing increasing levels of CO2 which are in turn causing the average temperature of the Earth to increase”. The awareness that the description in terms of causes and effects would dissolve if I decided to describe the Earth and the Sun in terms of the behaviour of their individual elementary particles*** doesn’t stop me from talking about causes and effects as long as I am describing things on a macroscopic level – any more than the awareness that ‘temperature’ is an emergent macroscopic property that does not exist on a more fundamental level stopped me from bursting out “Snow!? How can it be so cold in May?”

      If throwing out cause-and-effect as a fundamental concept led to large, obvious problems, we would have caught it. The reason I feel so confident that we can do without it is that a lot of thought has been put into it, and there really doesn’t seem to be anything that we want to do in science, that cannot be done without having cause-and-effect as a fundamental concept. It’s not that we particularly wanted to get rid of causality, it’s just that there’s been an increasing realisation that there doesn’t seem to be any real use for it. So the metaphor I’d use is not that we’re sawing off a branch we’re sitting on, but this: we’ve been working on an engine for a long time, rebuilding it again and again while trying to perfect it. We recently discovered that one piece of the engine that seemed very important when we started doesn’t actually seem to fit in anymore. We suppose we could bolt it on somewhere, but since the engine apparently runs just fine without it, it’s probably better just to put it aside. Maybe leave it within reach, if it turns out to be necessary after all, but for now there doesn’t seem to be any place for it.

      * For example, some people will say that momentum is conserved because the laws of physics are the same in different places**. Others will say the laws of physics are the same in different places because momentum is conserved. By Noether’s theorem the statements are equivalent, so you’re allowed to start out assuming either of them and then deriving the other.

      ** Ish. What we’re actually saying is that the Lagrangian is invariant under spatial translations.

      *** Which I wouldn’t. Even if it’s theoretically possible to describe a macroscopic system entirely in terms of it’s behaviour on a microscopic level, it’s practically never a good idea. Hence our built-in awareness that talking about causes and effects is a good practise in pretty much any situation we’ll normally encounter.


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