To be fair, there are basic things you do want kids to know, like how money works. The analogy given is imperfect because unlike music or art, sometimes math is just the means to an end. But it’s...
To be fair, there are basic things you do want kids to know, like how money works. The analogy given is imperfect because unlike music or art, sometimes math is just the means to an end. But it’s true that appreciation is important too.
As a computer programmer, I sometimes get frustrated with higher math because in the end I would like to write programs to do calculations. Earlier this year I picked up a book on geometry algebra that completely skipped anything about doing calculations because supposedly you could use a library written by someone else. (Eventually I concluded that for my purposes, quaternions are good enough.)
I think he addresses this pretty well. We want kids to be able to read nonfiction books for their education but we introduce them to fiction and poetry too.
The analogy given is imperfect because unlike music or art, sometimes math is just the means to an end.
I think he addresses this pretty well. We want kids to be able to read nonfiction books for their education but we introduce them to fiction and poetry too.
This seems like an over-broad claim to me? Many forms of thinking aren’t straightforwardly modeled using any kind of formal logic, because they aren’t about propositions that are unambiguously...
Any time you make some kind of assumption, and then try to reason your way to some kind of useful conclusion, you are doing math.
This seems like an over-broad claim to me? Many forms of thinking aren’t straightforwardly modeled using any kind of formal logic, because they aren’t about propositions that are unambiguously true or false, but rather something more nebulous. (See David Chapman on the rarity of absolute truths.) In Chapman’s terminology, I would say that most thinking is reasonable but not rational, because abstraction is a powerful but specialized tool.
As an aside, I do agree that the “programs are proofs” result is pretty neat, but have doubts about its importance. As I understand it, for a function to be an interesting proof, it needs to return a complicated type in a language specialized for proofs. There needs to be some doubt over whether any instance of that type exists, and usually there isn’t - most everyday types used in programming languages are trivially inhabited by constructing an instance directly. A function that returns an integer just proves that an integer exists, which we already knew.
That is, it proves it assuming the function returns at all. A useful language for proofs needs functions that are guaranteed to terminate, so you can show that the return value exists without running the program. But ordinarily we use languages that don’t guarantee termination and show that functions can terminate by running them and seeing that they do return an answer, since we care about what the answer is, not just that it exists.
On the contrary, Bayesian probability can be thought of as an extension of propositional logic (rather than a proposition being true/false, the proposition can take intermediary values). Since...
Many forms of thinking aren’t usefully modeled using any kind of formal logic, because they aren’t about propositions that are unambiguously true or false, but rather something more nebulous.
On the contrary, Bayesian probability can be thought of as an extension of propositional logic (rather than a proposition being true/false, the proposition can take intermediary values). Since most people have intuitive a priori beliefs, understanding Bayesian interference (at least in principle) can guide people to make better decisions.
For example, I've heard a few people express the belief that they had COVID-19 back in January, a proposition evidenced by some experienced general unwellness, usually accompanied by a fever and cough. However, despite these symptoms being compatible with COVID-19, relatively few people had the virus at that time, and therefore it's unlikely the malaise was actually COVID-19 (more likely, it was something else that presented similarly). In Bayesian parlance, we say the prior probability for having caught COVID-19 in January is low; consequently, the posterior probability of actually having caught COVID-19 in January should be appropriately penalized.
I think you’ve given a good example of a mathematical metaphor. There isn’t actual math here, just an intuitive judgement justified using mathematical concepts (like prior and posterior...
I think you’ve given a good example of a mathematical metaphor. There isn’t actual math here, just an intuitive judgement justified using mathematical concepts (like prior and posterior probability) with informal reasoning.
Even those of us who talk about Bayesian probability and priors a lot aren’t normally doing any probability calculations in our heads, or making a spreadsheet. If you don’t have any numbers to plug in, it doesn’t make sense to do this kind of calculation.
Contrast with a gambler playing poker or blackjack, where it actually makes sense to learn to count cards and the like. This works because games of chance are specifically designed to make probability work. But the real world is often more nebulous than a casino.
I think this is a good reason to learn something about the math, though, to understand such metaphors, even you rarely use it for calculations.
By “absolute truth” I meant something else; perhaps “unambiguous truth” would be a better way to put it. Conditional logic doesn’t really touch on the things that cause statements in everyday...
By “absolute truth” I meant something else; perhaps “unambiguous truth” would be a better way to put it. Conditional logic doesn’t really touch on the things that cause statements in everyday language to be somewhat nebulous. I’d encourage reading more of Chapman’s unfinished book. The first part seems to be close to done and does touch on multi-valued logic.
I agree that computer programming is mostly math and furthermore, ordinary programming is close to logic, with its if statements and boolean expressions. It’s a realm that was specifically constructed to allow logic to work. The difficulty comes from imperfectly modeling the real world. See falsehoods programmers believe about names for the kinds of issues that come up.
Machine learning is a somewhat successful way to handle nebulous data within a computer, and it’s all math too, but notice that logic isn’t very useful to describe how image recognition or machine translation happen.
To be fair, there are basic things you do want kids to know, like how money works. The analogy given is imperfect because unlike music or art, sometimes math is just the means to an end. But it’s true that appreciation is important too.
As a computer programmer, I sometimes get frustrated with higher math because in the end I would like to write programs to do calculations. Earlier this year I picked up a book on geometry algebra that completely skipped anything about doing calculations because supposedly you could use a library written by someone else. (Eventually I concluded that for my purposes, quaternions are good enough.)
I think he addresses this pretty well. We want kids to be able to read nonfiction books for their education but we introduce them to fiction and poetry too.
Sure, fair enough!
This seems like an over-broad claim to me? Many forms of thinking aren’t straightforwardly modeled using any kind of formal logic, because they aren’t about propositions that are unambiguously true or false, but rather something more nebulous. (See David Chapman on the rarity of absolute truths.) In Chapman’s terminology, I would say that most thinking is reasonable but not rational, because abstraction is a powerful but specialized tool.
As an aside, I do agree that the “programs are proofs” result is pretty neat, but have doubts about its importance. As I understand it, for a function to be an interesting proof, it needs to return a complicated type in a language specialized for proofs. There needs to be some doubt over whether any instance of that type exists, and usually there isn’t - most everyday types used in programming languages are trivially inhabited by constructing an instance directly. A function that returns an integer just proves that an integer exists, which we already knew.
That is, it proves it assuming the function returns at all. A useful language for proofs needs functions that are guaranteed to terminate, so you can show that the return value exists without running the program. But ordinarily we use languages that don’t guarantee termination and show that functions can terminate by running them and seeing that they do return an answer, since we care about what the answer is, not just that it exists.
On the contrary, Bayesian probability can be thought of as an extension of propositional logic (rather than a proposition being true/false, the proposition can take intermediary values). Since most people have intuitive a priori beliefs, understanding Bayesian interference (at least in principle) can guide people to make better decisions.
For example, I've heard a few people express the belief that they had COVID-19 back in January, a proposition evidenced by some experienced general unwellness, usually accompanied by a fever and cough. However, despite these symptoms being compatible with COVID-19, relatively few people had the virus at that time, and therefore it's unlikely the malaise was actually COVID-19 (more likely, it was something else that presented similarly). In Bayesian parlance, we say the prior probability for having caught COVID-19 in January is low; consequently, the posterior probability of actually having caught COVID-19 in January should be appropriately penalized.
For a more concrete example, see this.
For an example of how people misidentify their priors, see here.
I think you’ve given a good example of a mathematical metaphor. There isn’t actual math here, just an intuitive judgement justified using mathematical concepts (like prior and posterior probability) with informal reasoning.
Even those of us who talk about Bayesian probability and priors a lot aren’t normally doing any probability calculations in our heads, or making a spreadsheet. If you don’t have any numbers to plug in, it doesn’t make sense to do this kind of calculation.
Contrast with a gambler playing poker or blackjack, where it actually makes sense to learn to count cards and the like. This works because games of chance are specifically designed to make probability work. But the real world is often more nebulous than a casino.
I think this is a good reason to learn something about the math, though, to understand such metaphors, even you rarely use it for calculations.
By “absolute truth” I meant something else; perhaps “unambiguous truth” would be a better way to put it. Conditional logic doesn’t really touch on the things that cause statements in everyday language to be somewhat nebulous. I’d encourage reading more of Chapman’s unfinished book. The first part seems to be close to done and does touch on multi-valued logic.
I agree that computer programming is mostly math and furthermore, ordinary programming is close to logic, with its if statements and boolean expressions. It’s a realm that was specifically constructed to allow logic to work. The difficulty comes from imperfectly modeling the real world. See falsehoods programmers believe about names for the kinds of issues that come up.
Machine learning is a somewhat successful way to handle nebulous data within a computer, and it’s all math too, but notice that logic isn’t very useful to describe how image recognition or machine translation happen.