If someone could invent the ladder or dominoes explanation, that’d be a great sign about their conceptual understanding of induction. They could have weaknesses in their understanding, but at least something is going right. If they could merely repeat ladders or dominoes after it was told to them, then it might not mean much – it’s possible they understand it well or not. To find out, you’d need to ask them followup questions to try to get them to say some thoughts that aren’t repeating stuff they were told – try to get them to think on their feet / for themselves some and then say those thoughts.
I know that and am already taking it into account. I agree with what I think you mean here.
It wasn’t very clear, but it’s clearer after this additional message.
This is related to how, when you understand things well, it’s generally pretty easy to come up with new examples.
There are people who claim to understand things but then can’t give examples. This is very suspicious. It happens with philosophy sometimes – e.g. in response to my articles.
Happens in math+physics sometimes too. I agree it’s really bad.
There’s a lot of (edit: some) stuff in high energy theory / string theory that people think they know but then they can’t provide a good solid example of it working out.
In math, it seems like there’s a lot of higher category theory / derived algebraic geometry stuff that’s like that.
It’s hard to know for sure that these people are fos. In both cases the examples probably(?) exit (edit: exist), but they are extremely complicated and I think there are more who people act like they understand them than there are that really understand them.
I read multiple explanations of induction that say you can only induce from 1 (or 0) to positive infinity, and that induction always proves something for all natural numbers (which can be defined starting at 0 or 1).
Do you think you can induce from 50 to positive infinity? Is that a valid concept or an error?
I did already consider that even though you wrote
you might nevertheless answer “yes” to my question about inducing starting at 50. I didn’t think what you wrote there was a clear indication of what your conceptual understanding actually was.
It’s hard to tell sometimes when people are dumbing down explanations in problematic ways but their own concept is ok – it’s just the teaching that’s poor (or maybe just aimed at a target audience that doesn’t care about high quality conceptual understanding, just passing a few tests). Or sometimes they actually think about it how they wrote it. In this case, besides over-simplifying, they could also be over-complicating things by expecting you to map other sequences (like integers from 50 to positive infinity) to the natural numbers.
Yes, it doesn’t matter where you start in order to apply induction. If you started at 50 you’d have to prove the first 49 cases by hand (or however many, maybe you don’t even care about those, or maybe you care about every integer from -3 to 49 or something).
You could also induct over more interesting structures like the integers by doing something like: Showing that P(0) and then showing that (P(n) and P(-n)) => (P(n+1) and P(-n - 1) ) for all natural numbers n (incl 0 in the natural numbers ofc).
You could induct over an infinite tree by showing that P(node) => P(c) for every c that’s a child of the node.
etc
Oh that reminds me of perhaps the worst example of people having no examples: philosophical induction. I can basically never get people to even try to give worked, step-by-step examples of philosophical induction.
Deduction is also an issue. People can give simple deduction examples. But some people claim that complex arguments are equivalent to a bunch of deduction. But they can never give any example of translating a complex argument into deductions. You’d think if it was important and an actual thing people do, that someone would have written down an example in a book at least once, even if it was a big hassle. But these people cannot cite any example in addition to being unable to make one themselves. Sometimes they will take a complex argument and massively oversimplify it, and lose a lot of the content, while trying to turn it into deduction (and also usually their deduction is invalid and then they don’t understand the question when I ask them for a cite to a written, detailed specification of what they think the rules of deduction are). So that’s a bad, sloppy, ad hoc attempt at an example. And generally when questioned, and trying to defend it, they claim it’s just an approximation and not the full/real thing – so in other words it isn’t really a proper example, so they don’t actually have a real example.
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OK good. And you could also induce in the negative direction. Or you could induce upwards by increments of 2 for a claim about only even or odd numbers.
Re the first 49 cases, btw, my intent was just to skip them and not make any claim about them working.
Can you imagine that some other people might not think of it that way? They could have different concepts that are less convenient or flexible to work with or apply.
Okay. That’s really interesting.
I’m going to pay special attention to the “trivial” mistakes that I make when doing physics tomorrow, and I’ll see what I find.
Yes. I think I’ve seen it first-hand (maybe not with induction, but with similar things). People who just don’t “get it.” A lot of them even managed to get math or physics BScs, though in fairness they don’t seem to be the ones who are successful in academia.
Maybe people who get into grad school are better at hiding their basic conceptual errors, or even if they have fewer basic conceptual errors maybe they still have a significant amount that I can’t see. I’m now wondering if this goes some of the way towards explaining why the culture of academia is such that people (including me) are pretty afraid of asking questions in class and “looking dumb.”
Yeah that’s a factor.
So consider:
define f(x): 2x >= 100
The domain (allowed inputs) is integers.
So you can use 50 as a base case and inductively prove that f is true for the range {50…∞}.
But someone says “our process has to start at 1” or “we need proofs that apply to all natural numbers; give me one of those”. How can you make an inductive proof to satisfy them? It will not be perfect depending on the exact use case (since literally f is not true for e.g. 10), but there is something you can do that might solve their problem.
Also, can you imagine any scenarios where they are making a reasonable request?
There are certainly scenarios where it is not very reasonable. E.g. you get hired at the CIA and they say “our policy here at the CIA is we only record proofs about all natural numbers in our proofs database, cuz we thought if we put in just any proof we’d run out of disk space, so we needed a way to limit things”. Stupid government red tape… Nevertheless, you could do something to get a proof related to f into their database.
Answering this stuff helps connect induction to other concepts and also would demonstrate some flexible understanding that most high school students don’t have even if they are able to do some standard induction problems on their math tests. This is not meant to be very hard or long if you know the other relevant concepts I have in mind (which I think lmf does).
I have read material on several parts of physics. The material I have read on string theory is among the worst stuff I have read. The people who write about string theory don’t seem to be particularly interested in quantum theory or general relativity, or in explaining the problems they’re trying to solve. Their material reminds me of this quote from DD in a talk he gave called “Apart from universes” [1, 2 Chapter 18]:
There’s a related and even worse spanner in the works of elementary particle physics: particles (or fields, strings, or whatever) are supposed to be fully quantum-mechanical entities. But the people who work on them only ever construct classical, single-universe theories. Why? Because they think that the quantum part of the theory necessarily has to be trivial. It is assumed that in order to discover the true quantum-dynamical equations of the world, you have to enact a certain ritual. First you have to invent a theory that you know to be false, using a traditional formalism and laws that were refuted a century ago. Then you subject this theory to a formal process known as quantization (which for these purposes includes renormalization). And that’s supposed to be your quantum theory: a classical ghost in a tacked-on quantum shell.
define g(x) : f(x + 49),
then you can inductively prove g is true for {1,…,∞}
Well, I think stuff like that could be needed for formal proof assistants like coq. E.g. I think it would be completely reasonable to use the formal defn I gave (the one that only worked for predicates holding for all natural numbers) to define induction for the computer, then whenever you want to use it in a proof you have to modify your input predicate so that it fits.
I was going to define two new functions: one to map between sequences, and then one to compose the first two functions. But sure, your way works too.
Doing that shows how induction can be connected with other concepts (in my version, they are explicitly mapping and composing). Some people can do that and some can’t (or maybe only with guidance/hints/etc.). It’s a way people’s knowledge varies.
Similarly, some people find such things easy, intuitive, kinda fun. Others find it burdensome work to be avoided unless there is a clear purpose. The first group have a much better chance to accomplish stuff that’s creative.
Some people have rigid ideas like induction must work for all natural numbers. They might think you must use mapping and composing, or some other conversion method. But that’d be a bad way to conceptualize the issues. It’s good to be able to induce for 50+ directly without requiring extra steps, extra tacked on complexity which usually isn’t useful.
Make sense, agree?
This much I agree with. String theory people often don’t have great explanations of why they are solving the problems they are trying to solve, and even when they do they often don’t explain it clearly.
There’s a related and even worse spanner in the works of elementary particle physics: particles (or fields, strings, or whatever) are supposed to be fully quantum-mechanical entities. But the people who work on them only ever construct classical, single-universe theories. Why? Because they think that the quantum part of the theory necessarily has to be trivial.
Deutsch is just dead wrong about this.
I don’t know exactly what a “fully quantum mechanical” entity is, but I don’t think that the particles in QFT count (nor do I think strings or branes count, but that’s a more complicated discussion). Particles arise when you try to write your path integral as an asymptotic series in ħ, so the assumption that quantum effects are small is built into the whole formalism of perturbative QFT.
People who work on this stuff simply do not think that “the quantum part of the theory necessarily has to be trivial.” I thought that every theorist knows that this quasi-classical / perturbative way of studying QFT is not the whole story. Non-perturbative QFT is a huge area of study: almost everyone who calls themselves a string theorist works on some aspect of non-perturbative QFT, and “non-Lagrangian” field theories (i.e. quantum field theories with no classical limit) show up all the time.
The methods for studying non-perturbative QFT are more limited, so a lot of things about it are very mysterious. There’s not a lack of interest, though!
It is assumed that in order to discover the true quantum-dynamical equations of the world, you have to enact a certain ritual. First you have to invent a theory that you know to be false, using a traditional formalism and laws that were refuted a century ago. Then you subject this theory to a formal process known as quantization (which for these purposes includes renormalization). And that’s supposed to be your quantum theory: a classical ghost in a tacked-on quantum shell.
Since perturbative QFTs have classical limits, it’s not problematic at all that one starts with a classical field theory (there’s some extra wiggle room that comes up in one’s choice of renormalization scheme; different renormalization schemes would lead to different perturbative theories). Not every quantum theory can be obtained in this way, but I don’t think that theorists forget that.
Yes and yes.
Now that we clarified stuff more, want to try explaining trig more?
Okay.
It seems odd to me that you say you don’t understand something and then continue to write without asking questions. A longer explanation of what DD is talking about can be found here: