“Use your intuition to guide you, but try whenever there’s any doubt over what is correct to state what you mean in a rigorous form going back to axioms.”
Does this involve too many implicitly defined terms?
Who is this aimed at? Recreating Euclid? Then, yes, I think there are too many implicitly defined terms. Post-Euclid, who is helped by rigor? What advances are driven by rigor? The only one that springs to my mind is Gauss’s correct statement of the fundamental theorem of algebra.
I’m also not convinced that Euclid’s rigor mattered, but I don’t know how to assess this.
Post-Euclid, who is helped by rigor? What advances are driven by rigor? T
The primary advantages of rigor are that it helps prevents one from going down blind alleys or getting hopelessly confused. Thus for example in the 17th century much ink was spilled over what 1 −1 +1 −1 +1 −1… summed to. They didn’t have any rigorous notion of what the sum of an infinite series meant. And once this was answered the question essentially disappeared. Similarly, there were all sorts of apparent paradoxes and issues in calculus that were cleared up with delta-epsilon limit notions in the mid 19th century. And one can tell similar tales about early topology.
What happened in the 17th century? My understanding is that the rigorous definition of convergent series was given by Cauchy and Gregory’s (much greater) contribution was merely to hypothesize that only some series should be considered to have sums. This is an advance that I think is not so well connected to your suggestion. Even if he did give a rigorous definition, I doubt it made any reference to axioms. Was all this spilled ink a bad thing? Was it a blind alley full of the hopelessly confused? Or was the exploration necessary to accept the distinction or definition? It’s not like people stopped having opinions on that series.
As to your second and third examples, I don’t think they are examples of people spending much time in places that their successors labeled blind alleys.
My understanding is that the rigorous definition of convergent series was given by Cauchy and Gregory’s (much greater) contribution was merely to hypothesize that only some series should be considered to have sums. This is an advance that I think is not so well connected to your suggestion.
Well, no, because it turns out that you have multiple notions of what it means to take the sum of the series. Abel summation would be one example. That said, I agree that not everything should be summable is a major insight, but even this came about as part of Cauchy et. al.’s attempt to make things rigorou.
As to your second and third examples, I don’t think they are examples of people spending much time in places that their successors labeled blind alleys.
Sure, aspects of their work ended up being useful, but other times books and papers had results that relied on “theorems” that simply weren’t true. For example, many took for granted that a continuous function had to be differentiable almost everywhere until Weierstrauss gave counterexamples.
When are you claiming all this work on 1-1+1… ended? in the 17th century as in your original comment, or with Cauchy? Do you really dispute that Gregory in the 17th century claimed that not all series should be summable?
Sure, aspects of their work ended up being useful, but other times books and papers had results that relied on “theorems” that simply weren’t true.
You seem to be saying that sloppy proofs of true theorems are not useful.
Math:
“Use your intuition to guide you, but try whenever there’s any doubt over what is correct to state what you mean in a rigorous form going back to axioms.”
Does this involve too many implicitly defined terms?
Who is this aimed at? Recreating Euclid? Then, yes, I think there are too many implicitly defined terms. Post-Euclid, who is helped by rigor? What advances are driven by rigor? The only one that springs to my mind is Gauss’s correct statement of the fundamental theorem of algebra.
I’m also not convinced that Euclid’s rigor mattered, but I don’t know how to assess this.
The primary advantages of rigor are that it helps prevents one from going down blind alleys or getting hopelessly confused. Thus for example in the 17th century much ink was spilled over what 1 −1 +1 −1 +1 −1… summed to. They didn’t have any rigorous notion of what the sum of an infinite series meant. And once this was answered the question essentially disappeared. Similarly, there were all sorts of apparent paradoxes and issues in calculus that were cleared up with delta-epsilon limit notions in the mid 19th century. And one can tell similar tales about early topology.
What happened in the 17th century? My understanding is that the rigorous definition of convergent series was given by Cauchy and Gregory’s (much greater) contribution was merely to hypothesize that only some series should be considered to have sums. This is an advance that I think is not so well connected to your suggestion. Even if he did give a rigorous definition, I doubt it made any reference to axioms. Was all this spilled ink a bad thing? Was it a blind alley full of the hopelessly confused? Or was the exploration necessary to accept the distinction or definition? It’s not like people stopped having opinions on that series.
As to your second and third examples, I don’t think they are examples of people spending much time in places that their successors labeled blind alleys.
Well, no, because it turns out that you have multiple notions of what it means to take the sum of the series. Abel summation would be one example. That said, I agree that not everything should be summable is a major insight, but even this came about as part of Cauchy et. al.’s attempt to make things rigorou.
Sure, aspects of their work ended up being useful, but other times books and papers had results that relied on “theorems” that simply weren’t true. For example, many took for granted that a continuous function had to be differentiable almost everywhere until Weierstrauss gave counterexamples.
When are you claiming all this work on 1-1+1… ended? in the 17th century as in your original comment, or with Cauchy? Do you really dispute that Gregory in the 17th century claimed that not all series should be summable?
You seem to be saying that sloppy proofs of true theorems are not useful.