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A_Seagull wrote:Philosophically speaking, '1+1=2' is a string of 5 symbols that is declared to be 'true' within the system of 'mathematics'.
If you were to create your own system call it 'Raminatics' if you like, then you could set up symbols and axioms and means of generating theorems from that system which would then be 'true' within that system.
Ramin22 wrote:Thank you everyone for the answers. I have one more question. Is 'either P is true or ~P is true' just true inside classical logic? I know it can be considered true just in that sense. But is there any other sense, or are there others who argue classical logic is true on other grounds?
Alan Masterman wrote:A most amusing and interesting thread. Very few of the posts have any connection with the question, but that is quite normal and predictable, and (in this case) forgivable because the question is a question in mathematical philosophy, a subject of which very few mathematicians have any understanding. .
You: Suppose you’re in the garden and you see two worms crawling around. Then two more worms crawl over. How many worms do you have now?
Peter: “Crawling” means moving around on your hands and knees. Worms don’t have hands and knees, so they don’t “crawl.” They have hair-like projections called setae which make contact with the soil, and their bodies are moved by two sets of muscles, an outer layer called the circular muscles and an inner layer known as the longitudinal muscles. Alternation between these muscles causes a series of expansions and contractions of the worm’s body.
You: That’s all very impressive, but you know what I meant, Peter, and the specific way worms move around is completely irrelevant in any case. The point is that you’d have four worms.
Peter: Science is irrelevant, huh? Well, do you drive a car? Use a cell phone? Go to the doctor? Science made all that possible.
You: Yes, fine, but what does that have to do with the subject at hand? What I mean is that how worms move is irrelevant to how many worms you’d have in the example. You’d have four worms. That’s true whatever science ends up telling us about worms.
Peter: You obviously don’t know anything about science. If you divide a planarian flatworm, it will grow into two new individual flatworms. So, if that’s the kind of worm we’re talking about, then if you have two worms and then add two more, you might end up with five worms, or even more than five. So much for this a priori “arithmetic” stuff.
You: That’s a ridiculous argument! If you’ve got only two worms and add another two worms, that gives you four worms, period. That one of those worms might later go on to be divided in two doesn’t change that!
Peter: Are you denying the empirical evidence about how flatworms divide?
You: Of course not. I’m saying that that empirical evidence simply doesn’t show what you think it does.
Peter: This is well-confirmed science. What motivation could you possibly have for rejecting what we know about the planarian flatworm, apart from a desperate attempt to avoid falsification of your precious “arithmetic”?
You: Peter, I think you might need a hearing aid. I just got done saying that I don’t reject it. I’m saying that it has no bearing one way or the other on this particular question of whether two and two make four. Whether we’re counting planarian flatworms or Planters peanuts is completely irrelevant.
Peter: So arithmetic is unfalsifiable. Unlike scientific claims, for which you can give rational arguments.
You: That’s a false choice. The whole point is that argumentation of the sort that characterizes empirical science is not the only kind of rational argumentation. For example, if I can show by reductio ad absurdum that your denial of some claim of arithmetic is false, then I’ve given a rational justification of that claim.
Peter: No, because you haven’t offered any empirical evidence.
You: You’ve just blatantly begged the question! Whether all rational argumentation involves the mustering of empirical evidence is precisely what’s at issue.
Peter: So you say now. But earlier you gave the worm example as an argument for the claim that two and two make four. You appeal to empirical evidence when it suits you and then retreat into unfalsifiability when that evidence goes against you.
You: You completely misunderstand the nature of arithmetical claims. They’re not empirical claims in the same sense that claims about flatworm physiology are. But that doesn’t mean that they have no relevance to the empirical world. Given that it’s a necessary truth that two and two make four, naturally you are going to find that when you observe two worms crawl up beside two other worms, there will be four worms there. But that’s not “empirical evidence” in the sense that laboratory results are empirical evidence. It’s rather an illustration of something that is going to be the case whatever the specific empirical facts turn out to be.
Peter: See, every time I call attention to the scientific evidence that refutes your silly “arithmetic,” you claim that I “just don’t understand” it. Well, I understand it well enough. It’s all about trying to figure out flatworms and other things science tells us about, but by appealing to intuitions or word games about “necessary truth” or just making stuff up. It’s imaginary science. What we need is real, empirical science, like physics.
You: That makes no sense at all. Physics presupposes arithmetic! How the hell do you think physicists do their calculations?
Peter: Whatever. Because science. Because I @#$%&*! love science.
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