I am getting more and more sympathetic to Dr. Wildberger's views on modern mathematics and it makes me want to look back at the criticisms already being put forward as set theory was being developed. The fact that he has come forward with these ideas and criticisms is especially interesting for two reasons: 1) Professor Wildberger comes from a position of competence in modern analysis and an apparently solid familiarity with the history. 2) He has already made progress in developing interesting and apparently original mathematics, esp. geometry, in terms that, as far as I can tell, do not depend on the usual machinery of infinite sets (though I wonder if an explicit formal theory of types or some such thing will need to be employed or if, more likely, the involvement of formal logic is unnecessary due to the transparent simplicity of the assumptions and reasoning). Watch his rational trig series. The irony is that it is hard to see how someone who has not spent a good deal of time gaining competence in modern mathematics and some inkling of troubles with foundations, could fully appreciate the significance of what he is trying to do. At the same time, that familiarity itself might also be the very thing that forms an obstacle (Its giving me troubles!). In any case, I'm now officially cheering on Professor Wildberger. I could still articulate criticisms I suppose. It is just that those are being overshadowed by the promise of his his rational numbers approach. It might be profound. Let's see. I am ordering his book and forming a small study group of mathematicians to discuss it. It should be noted by way of full disclosure, that I am a differential geometer and not a logician, number theorist or an expert in math foundations (yet). Finally, I might add a couple caveats: I am inclined to say that there is, perhaps such an idea as the infinite and we should be in awe. Cantor was an important figure whose ideas have a certain undeniable clarity and his theory certainly has structure; mathematicians and maths grad students pretty well know how to play that game reliably so it isn't complete mush. For me what is mostly in doubt is the idea that the notion of the infinite can be tamed by the notions of set theory *while at the same time* being a theory that is more than a game. The question is whether formal set theory is really consequentially and meaningfully tethered to the rather solid reality we know as arithmetic (money, cattle, and computers). By "meaningfully tethered" I mean something like being connected to the "empirical" reality of arithmetic in a way that really compels logically and has explanatory powers. I used to think ZFC did just that, but now I am far less sure. Of course, a certain skeptical mood had been growing in me for some time.
I am getting more and more sympathetic to Dr. Wildberger's views on modern mathematics and it makes me want to look back at the criticisms already being put forward as set theory was being developed. The fact that he has come forward with these ideas and criticisms is especially interesting for two reasons: 1) Professor Wildberger comes from a position of competence in modern analysis and an apparently solid familiarity with the history. 2) He has already made progress in developing interesting and apparently original mathematics, esp. geometry, in terms that, as far as I can tell, do not depend on the usual machinery of infinite sets (though I wonder if an explicit formal theory of types or some such thing will need to be employed or if, more likely, the involvement of formal logic is unnecessary due to the transparent simplicity of the assumptions and reasoning). Watch his rational trig series. The irony is that it is hard to see how someone who has not spent a good deal of time gaining competence in modern mathematics and some inkling of troubles with foundations, could fully appreciate the significance of what he is trying to do. At the same time, that familiarity itself might also be the very thing that forms an obstacle (Its giving me troubles!). In any case, I'm now officially cheering on Professor Wildberger. I could still articulate criticisms I suppose. It is just that those are being overshadowed by the promise of his his rational numbers approach. It might be profound. Let's see. I am ordering his book and forming a small study group of mathematicians to discuss it. It should be noted by way of full disclosure, that I am a differential geometer and not a logician, number theorist or an expert in math foundations (yet).
Finally, I might add a couple caveats: I am inclined to say that there is, perhaps such an idea as the infinite and we should be in awe. Cantor was an important figure whose ideas have a certain undeniable clarity and his theory certainly has structure; mathematicians and maths grad students pretty well know how to play that game reliably so it isn't complete mush. For me what is mostly in doubt is the idea that the notion of the infinite can be tamed by the notions of set theory *while at the same time* being a theory that is more than a game. The question is whether formal set theory is really consequentially and meaningfully tethered to the rather solid reality we know as arithmetic (money, cattle, and computers). By "meaningfully tethered" I mean something like being connected to the "empirical" reality of arithmetic in a way that really compels logically and has explanatory powers. I used to think ZFC did just that, but now I am far less sure. Of course, a certain skeptical mood had been growing in me for some time.
Thanks. And eagerly waiting for your next Video