21–30 of 44 posts
Earlier quoted context omitted.
Not really, because having one something plus one something equal two somethings is pretty much the definition of discrete object.
Yes, I agree and think this is a very smart observation. It doesn't hold for jugs of water, say, if you're allowed to pour water between the jugs.
Cool article, though.
Earlier quoted context omitted.
Not really, because having one something plus one something equal two somethings is pretty much the definition of discrete object.
Yes, I agree and think this is a very smart observation. It doesn't hold for jugs of water, say, if you're allowed to pour water between the jugs.
Some math was designed to be useful.
Science is by definition those practical problems to which math can be applied.
He still doesn't know.
Very disappointing.
In fact "simple" mathematics are not simple at all by any objective measure. Starting with any truly formal system, you need a stupendous number of deductions to get to things like elementary laws of arithmetic, or basic plane geometry. Mathematical proofs are not formal proofs--they are instructions for our brains. Evolution made the relevant parts of our brains the same, so same instructions lead to same results. That's why there's never any argument over whether a proof is correct, once a few people got to study it in detail. This also explains Hamming's observation that when proofs turn out to be "wrong" after math has evolved a bit, theorems are still usually correct. We find a new, better route to the same place in our brain, and recognize the hazards of the old route, now deprecated.
Okay, here is the key bit: if evolution made the relevant parts of our brains the same, that means it has arrived at a maximum, or at least a local maximum. What is the nature of this maximum? Physiologically, there are constraints on the amount of brain circuity our body can maintain. Brains consume a lot of energy, take up space, etc. So naturally, evolution ended up with a design where the same circuity can serve the greatest possible number of functions.
Of course, evolution only concerns itself with those functions relevant to our survival and reproduction. But there is nothing niche about those goals. If some general pattern occurs often in our quest for survival, then it likely occurs often in other quests that evolution never knew about--like building airplanes.
This guy could easily cut 50% of his words and lose no meaning. He needs an editor.
I think this kind of intent requires some seemingly superfluous amount of wording to carry a nontrivial flow of narrative to be persuasive if to avoid dullness in expression.
Earlier quoted context omitted.
Yes, I agree and think this is a very smart observation. It doesn't hold for jugs of water, say, if you're allowed to pour water between the jugs.
It doesn't hold for loads of things. 1 heap + 1 heap = 1 heap. 1 bunny + 1 bunny = 12 bunnies, depending on how fast you are with your observing. 1 dl of ethanol + 10 dl of water < 11 dl of diluted alcohol.
Mathematics is a product of the mind; the mind is a product of evolution; evolution is a product of natural laws. If you want answers to these questions, it helps to look at what we are, exactly, and how we came to be. In fact "simple" mathematics are not simple at all by any objective measure. Starting with any truly formal system, you need a stupendous number of deductions to get to things like elementary laws of a…
old proofs of theorems may become false proofs. The old proofs no longer cover the newly defined things. The miracle is that almost always the theorems are still true; it is merely a matter of fixing up the proofs. It is claimed that an ex-editor of Mathematical Reviews once said that over half of the new theorems published these days are essentially true though the published proofs are false. I'm only around 55% of…
Mathematics is a product of the mind; the mind is a product of evolution; evolution is a product of natural laws. If you want answers to these questions, it helps to look at what we are, exactly, and how we came to be. In fact "simple" mathematics are not simple at all by any objective measure. Starting with any truly formal system, you need a stupendous number of deductions to get to things like elementary laws of a…
Mathematics is not a product of the mind any more than physics is. All theorems are true (or, more precisely, all theorems follow from their axioms), even the ones we haven't discovered yet. That a mind can choose an axiomatic system to explore does not mean the relationships between those axioms and their theorems are created by that mind.
Earlier quoted context omitted.
Hm, it's not productive to argue over analogies. They're just used to illustrate a point. I was saying math was explicitly designed to work & to be general. Likewise words were designed for communication (not just for describing the real world), & I'd say that sentence communicates your meaning very well. eg later he says: >" Is it not remarkable that 6 sheep plus 7 sheep make 13 sheep; that 6 stones plus 7 stones ma…
It's right there on the top: "It is evident from the title that this is a philosophical discussion". In case you aren't interested in philosophy, I don't see why bother reading the article and commenting on it -- unless perhaps you want to say something against doing any philosophy in the first place. But if you are interested in philosophy, then you should know that the philosopher often starts by wondering about so…
Counting is defined by objects, and 13 is the sum of 6 and 7. Indeed it is an empirical fact in as much as it has a physical interpretation.