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user21349

I don't think Dedekind's complaint can be that a proof in an axiomatic system must be invalid if the objects in the system hadhave not been constructed from simpler objects. This obviously leads to an infinite regress. Today we typically take "set" to be undefined, but Dedekind would surely have been familiar with treatments of Euclidean geometry in which "point" is an undefined term.

I suspect that DedekingDedekind was complaining that people were not working in a full axiomatic system for the reals, but rather were using only the elementary axioms and did not realize that the completeness axiom was needed. Note that in the quote, the argument he criticizes uses only elementary properties such as commutativity and associativity.

Without completeness, we don't have any way of proving the existence of square roots, but the textbook treatments he complains about presumably don't admit that the theorem $\sqrt{2}\sqrt{3}=\sqrt{6}$ is conditional on the unproved existence of these roots.

Why couldn't it have been proved by Euclid?

Euclid couldn't have proved it to modern standards of rigor because Euclid couldn't have proved that the circles in your diagram intersected. This is equivalent to lacking the completeness property of the reals.

I don't think Dedekind's complaint can be that a proof in an axiomatic system must be invalid if the objects in the system had not been constructed from simpler objects. This obviously leads to an infinite regress. Today we typically take "set" to be undefined, but Dedekind would surely have been familiar with treatments of Euclidean geometry in which "point" is an undefined term.

I suspect that Dedeking was complaining that people were not working in a full axiomatic system for the reals, but rather were using only the elementary axioms and did not realize that the completeness axiom was needed. Note that in the quote, the argument he criticizes uses only elementary properties such as commutativity and associativity.

Without completeness, we don't have any way of proving the existence of square roots, but the textbook treatments he complains about presumably don't admit that the theorem $\sqrt{2}\sqrt{3}=\sqrt{6}$ is conditional on the unproved existence of these roots.

Why couldn't it have been proved by Euclid?

Euclid couldn't have proved it to modern standards of rigor because Euclid couldn't have proved that the circles in your diagram intersected. This is equivalent to lacking the completeness property of the reals.

I don't think Dedekind's complaint can be that a proof in an axiomatic system must be invalid if the objects in the system have not been constructed from simpler objects. This obviously leads to an infinite regress. Today we typically take "set" to be undefined, but Dedekind would surely have been familiar with treatments of Euclidean geometry in which "point" is an undefined term.

I suspect that Dedekind was complaining that people were not working in a full axiomatic system for the reals, but rather were using only the elementary axioms and did not realize that the completeness axiom was needed. Note that in the quote, the argument he criticizes uses only elementary properties such as commutativity and associativity.

Without completeness, we don't have any way of proving the existence of square roots, but the textbook treatments he complains about presumably don't admit that the theorem $\sqrt{2}\sqrt{3}=\sqrt{6}$ is conditional on the unproved existence of these roots.

Why couldn't it have been proved by Euclid?

Euclid couldn't have proved it to modern standards of rigor because Euclid couldn't have proved that the circles in your diagram intersected. This is equivalent to lacking the completeness property of the reals.

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user21349
user21349

I don't think Dedekind's complaint can be that a proof in an axiomatic system must be invalid if the objects in the system had not been constructed from simpler objects. This obviously leads to an infinite regress. Today we typically take "set" to be undefined, but Dedekind would surely have been familiar with treatments of Euclidean geometry in which "point" is an undefined term.

I suspect that Dedeking was complaining that people were not working in a full axiomatic system for the reals, but rather were using only the elementary axioms and did not realize that the completeness axiom was needed. Note that in the quote, the argument he criticizes uses only elementary properties such as commutativity and associativity.

Without completeness, we don't have any way of proving the existence of square roots, but the textbook treatments he complains about presumably don't admit that the theorem $\sqrt{2}\sqrt{3}=\sqrt{6}$ is conditional on the unproved existence of these roots.

Why couldn't it have been proved by Euclid?

Euclid couldn't have proved it to modern standards of rigor because Euclid couldn't have proved that the circles in your diagram intersected. This is equivalent to lacking the completeness property of the reals.