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I'm really pleased that you've been willing to stick your head above the parapet and say this. What you've said seems to me to be a common line of reasoning, b

I'm really pleased that you've been willing to stick your head above the parapet and say this. What you've said seems to me to be a common line of reasoning, but it always confuses me. In particular, you say: > Sadly, I'm not math savvy. OK, that's cool, and I'm always willing to engage with people who want to learn. > Obviously (x^x infinitum) is not the same as (x^x infinitum-1). Given that you're not math savvy, that's a pretty definite thing to say. > ... my answer is based on logical reasoning rather than mathematical proof. Well, actually your answer seems to be based on intuition, and not on reasoning at all. You intuitively say that x^x^x^x != x^x^x, which is fair enough, and then leap to the conclusion that the infinite case must be like the finite case. And there's a problem. The infinite case is not like the finite case. Similarly, the infinite sum 1/2+1/4+1/8+1/16+1/32+... insofar as it can be given any meaning at all, must be set equal to 1, even though all the finite partial sums are strictly less than 1. > this doesn't seem like a very interesting "math paradox". But your "reasoning" is wrong. Perhaps you've dismissed it as "uninteresting" because you haven't really understood it? That's fair enough - I don't have a problem with that, but it would be nice to believe that you realised it. Let me just say again that I'm really pleased you expressed your thoughts on this. I'm sure you're not the only one, and I find it useful to hear the way people think about these things. I give lots of presentations and masterclasses on math to all ranges of ages and experience. I need to know the intuitions people are using. So thanks.

giberson 16y ago
Thanks for your response Rider, and let me re-iterate that I'm already aware my reasoning is wrong. Hence my allusion to how I think Spock would reply to me. For clarification, I'm considering my observation that x^x infinitum is not the same as x^x infinitum-1 as based on reason rather than intuition because there is reasoning behind my statement. This may be an argument of semantic and thats certainly what I want to avoid since it's useless. But my answer isn't based on an instinctive knowing (intuition) it was based on the following considerations: 3^1 = 3 3^2 = 9 3^3 = 27 For any n^x I could* think of, n^(x-1) was a different value. If I think of infinity as some unbounded large number, then I think of x-1 as a smaller (by 1) unbounded large number. It may be fair to consider this an intuition, but since I'm basing it off of observations of smaller number examples I think its closer to an implication of the previous observed behavior. With that argument I contest that my conclusion is in fact based on reason. However, I will and still do submit that my reasoning is that of a layman and may be (and probably is) incorrect. But, my conclusion seems to be inline with that of the other comments to this thread regarding the overall evaluation 2=4 through this number game. Now, I'm making an assumption here and by all rights perhaps making an ass of myself, but from my understanding of the majority responses is an agreement that 2=x^x(infinitum) does not lead to a conclusion of 2 != 4. Their arguments are more technical and perhaps outside my understanding level, but I think this is where my point lies about the math paradox being interesting. I would think that the paradox would only be interesting if the mathematical reasoning would be counter intuitive than a layman's intuition. That is to say, if the layman says it should be yes, mathematically it should be no in order for it to be interesting. (Again, I'm making the assumption original post argument doesn't hold water based on the refutation of the comment thread) Thanks for taking the time to reply, level headed even without being insulting, to my [mathematically incompetent] comments. :) * while formulating this reply, it occurred to me that if n = 1, 1^x = 1^(x-1). Here during this argument I've discovered a flaw in my reasoning. Which I am conceding in my foot note as my comment above is meant to explain my original reasoning.

giberson 16y ago

Thanks for your response Rider, and let me re-iterate that I'm already aware my reasoning is wrong. Hence my allusion to how I think Spock would reply to me. For clarification, I'm considering my observation that x^x infinitum is not the same as x^x infinitum-1 as based on reason rather than intuition because there is reasoning behind my statement. This may be an argument of semantic and thats certainly what I want to avoid since it's useless. But my answer isn't based on an instinctive knowing (intuition) it was based on the following considerations: 3^1 = 3 3^2 = 9 3^3 = 27 For any n^x I could* think of, n^(x-1) was a different value. If I think of infinity as some unbounded large number, then I think of x-1 as a smaller (by 1) unbounded large number. It may be fair to consider this an intuition, but since I'm basing it off of observations of smaller number examples I think its closer to an implication of the previous observed behavior. With that argument I contest that my conclusion is in fact based on reason. However, I will and still do submit that my reasoning is that of a layman and may be (and probably is) incorrect. But, my conclusion seems to be inline with that of the other comments to this thread regarding the overall evaluation 2=4 through this number game. Now, I'm making an assumption here and by all rights perhaps making an ass of myself, but from my understanding of the majority responses is an agreement that 2=x^x(infinitum) does not lead to a conclusion of 2 != 4. Their arguments are more technical and perhaps outside my understanding level, but I think this is where my point lies about the math paradox being interesting. I would think that the paradox would only be interesting if the mathematical reasoning would be counter intuitive than a layman's intuition. That is to say, if the layman says it should be yes, mathematically it should be no in order for it to be interesting. (Again, I'm making the assumption original post argument doesn't hold water based on the refutation of the comment thread) Thanks for taking the time to reply, level headed even without being insulting, to my [mathematically incompetent] comments. :) * while formulating this reply, it occurred to me that if n = 1, 1^x = 1^(x-1). Here during this argument I've discovered a flaw in my reasoning. Which I am conceding in my foot note as my comment above is meant to explain my original reasoning.