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Misinterpretations of intellectual history can arise from a misunderstanding of the strengths of the present.
"We are great because of X. We're superior to the ancients. Therefore the ancients lacked X."
If X is wrong, you may underestimate the past.
A thread... https://twitter.com/SamoBurja/status/1338121322874933248
"We are great because of X. We're superior to the ancients. Therefore the ancients lacked X."
If X is wrong, you may underestimate the past.
A thread... https://twitter.com/SamoBurja/status/1338121322874933248
For the claim that the Greeks didn't understand infinity or zero, the order might be:
"Our strength is precise concepts. We're better than the ancient Greeks. Therefore they lacked precise concepts."
Then simplify that for popular audiences, and voilà!
"Our strength is precise concepts. We're better than the ancient Greeks. Therefore they lacked precise concepts."
Then simplify that for popular audiences, and voilà!
But, you may ask, how does this misunderstand our strengths? Isn't it true that we have much more precise concepts than the ancient Greeks?
Don't the Peano axioms encode a clearer understanding of zero? Didn't Cantor give us a better understanding of infinity?
Don't the Peano axioms encode a clearer understanding of zero? Didn't Cantor give us a better understanding of infinity?
The answer is that we should distinguish two types of clarity: formal clarity and conceptual clarity.
Formal clarity involves having all of the elements of your formal system be specified.
Conceptual clarity involves understanding what is in the concepts you're thinking with.
Formal clarity involves having all of the elements of your formal system be specified.
Conceptual clarity involves understanding what is in the concepts you're thinking with.
Modern thought is far superior to ancient Greek thought with respect to formal clarity.
The Peano axioms encode a *formally* clearer sense of zero, and Cantor gave definitions that encode a *formally* clearer understanding of infinity.
The Peano axioms encode a *formally* clearer sense of zero, and Cantor gave definitions that encode a *formally* clearer understanding of infinity.
The rise of formal systems coincided with a movement away from aiming for conceptual clarity, and even from believing that conceptual clarity is a legitimate thing to aim for.
Nevertheless, it is clear that the Greeks were focused on conceptual clarity.
Nevertheless, it is clear that the Greeks were focused on conceptual clarity.
A focus on formal clarity yields a different set of questions and acceptable answers than a focus on conceptual clarity. The patterns of inquiry are different.
Here are two illustrations of the differences, one with respect to zero, the other with respect to infinity:
Here are two illustrations of the differences, one with respect to zero, the other with respect to infinity:
Formal clarity vs. conceptual clarity, on zero:
"What is zero? Is zero real? Is it not nothing? Is nothing a thing? Clearly not!"
Greeks:
Us: We have defined "0" in our formal system. Nothing to see here folks!
(Greeks: But are all nothings the same?)
(Us:
)
"What is zero? Is zero real? Is it not nothing? Is nothing a thing? Clearly not!"
Greeks:
Us: We have defined "0" in our formal system. Nothing to see here folks!
(Greeks: But are all nothings the same?)
(Us:
)
Formal clarity vs. conceptual clarity, on infinity:
"The natural numbers (0, 1, 2, 3...) and the integers (..., -2, -1, 0, 1, 2, ...) can be put into one-to-one correspondence. Yet the former are a proper subset of the latter. Are there more integers than natural numbers?"
—>
"The natural numbers (0, 1, 2, 3...) and the integers (..., -2, -1, 0, 1, 2, ...) can be put into one-to-one correspondence. Yet the former are a proper subset of the latter. Are there more integers than natural numbers?"
—>
(cont.)
Us: Our definition of "same number as" says that two sets that can be put into one-to-one correspondence have the "same number" of elements. Integers and natural numbers can be. So, yes.
Greeks: ...these are actually existing completed infinities?
—>
Us: Our definition of "same number as" says that two sets that can be put into one-to-one correspondence have the "same number" of elements. Integers and natural numbers can be. So, yes.
Greeks:
...these are actually existing completed infinities?—>
(cont.)
Us: 'Completed' infinity is not a thing. We have formally defined our system. Look at the cool results! They are cool, you have to admit.
Greeks: ...but I am not sure you are uniting flax with flax, as the proverb has it.
(Us:)
Us: 'Completed' infinity is not a thing. We have formally defined our system. Look at the cool results! They are cool, you have to admit.
Greeks:
...but I am not sure you are uniting flax with flax, as the proverb has it.(Us:
)
Tying this all together...
We have the strength of the precision that arises from formal clarity. We mistake that for *precision in general*. This can lead to an underestimation of the quality of ancient Greek thought.
We have the strength of the precision that arises from formal clarity. We mistake that for *precision in general*. This can lead to an underestimation of the quality of ancient Greek thought.
I'm not actually sure if "the Greeks didn't understand zero or infinity" arose as I've described. It's one possible story. (I'd be interested in other plausible stories.)
But I do think the above is a good illustration of a way in which people can come to underestimate the past.
But I do think the above is a good illustration of a way in which people can come to underestimate the past.
The real question is whether "conceptual clarity," i.e., trying to understand precisely what is actually in our own concepts, is a sensible and important goal.
If not, progress has been straightforward.
If so, then while we've gained something, we've also lost something.
If not, progress has been straightforward.
If so, then while we've gained something, we've also lost something.
