Image Credit & Copyright: Rolf Geissinger
"According to Holder one of the essential characteristics of the mathematical method can be described as building new notions as superstructure to notions present at a certain stage, in the following sense. The notions and methods applied at a certain stage are envisaged as objects of the mathematical investigation at a higher stage. For instance: one applies a certain algorithm or method of proof, and afterwords one considers the scope and the limits of the method, making the method itself an object of investigation. From this, Holder concludes that it is impossible to comprehend the whole of mathematics by means of a logical formalism, because logical considerations concerning the scope and limits of the formalism necessarily transcend the formalism and yet belong to mathematics." - Otto Holder(through Van Der Waerden).
This quote is more Van Der Waerden than Otto Holder; so, I don't know how much to really comment on it. The original thoughts are in German. There's certainly some truth to it. For instance, at one time number was a whole structure of counting method. Next, it's abstracted and used as an object in a higher mathematical universe(that of algebra). I'm thinking that these methods are like Jacob Bronowski's "inferred units"(in his Origins of Knowledge and Imagination). Jacob is always pointing out that the relations of mathematical concepts are 'action verbs" and not 'is' statements. But, there is equality in mathematics as well.
As for the last part about mathematics cannot be encompassed in logic. That depends on what you mean by logic. Not to mention what one values. If one values a finite set of axioms that can prove an infinity of truths, then no, logic cannot encompass mathematics. But, if one likes an open consistent mathematics, then a little perspective suggests that logic has it's place, although one changed from an Euclidean, or even a Pythagorean viewpoint of symbolic logic.
This quote is more Van Der Waerden than Otto Holder; so, I don't know how much to really comment on it. The original thoughts are in German. There's certainly some truth to it. For instance, at one time number was a whole structure of counting method. Next, it's abstracted and used as an object in a higher mathematical universe(that of algebra). I'm thinking that these methods are like Jacob Bronowski's "inferred units"(in his Origins of Knowledge and Imagination). Jacob is always pointing out that the relations of mathematical concepts are 'action verbs" and not 'is' statements. But, there is equality in mathematics as well.
As for the last part about mathematics cannot be encompassed in logic. That depends on what you mean by logic. Not to mention what one values. If one values a finite set of axioms that can prove an infinity of truths, then no, logic cannot encompass mathematics. But, if one likes an open consistent mathematics, then a little perspective suggests that logic has it's place, although one changed from an Euclidean, or even a Pythagorean viewpoint of symbolic logic.
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