What evidence do you have that -2+1=-1?
There are some things we “know” other than through direct evidence.
Which is good since there are many hard (impossible?) to answer criticisms of the idea that we can gain direct evidence of the ‘real world’ at all.
Math itself is just arbitrary rules like chess. You might as well ask “What evidence do we have that bishops move diagonally?”. That’s just the rule we’ve decided on. We could equally say they move like knights instead.
That’s why math itself cannot tell us anything about the real world. Evidence in math flows in the other direction: We choose the rules in math in order to model what we know about the real world. Or just to play with a cool abstract rule set. Although sometimes the cool abstract rule sets (like various forms of non-euclidean geometry) turn out to have applications in the real world. But those are just happy accidents.
What do you make of probability theory? Are we observing evidence through trial and error or are we deciding rules to model what we have observed? What about astronomy? Does it really matter who decides the rules?
Probably theory was created by gamblers like Gerolamo Cardano who wanted to know why they were losing money, and the first applications of it were to model dice. Obviously it has applications beyond gambling, though, but that was the first and most obvious application.
I’m not sure where you are going with astronomy. That’s just a normal science – people observed celestial bodies and made models (both physical and mathematical) to predict the motions. Of course, many early models incorrectly assumed the Earth was the center, but with relative motions that doesn’t matter. It does bring up the issue of to the degree that our models reflect the physical nature of the universe, but as with astronomy eventually the observational evidence invalidated the geocentric universe.
I don’t know what you think faith and gut feeling are. They aren’t the product of random number generators. I think it’s a little absurd to think that gut feelings didn’t keep up alive through most of our prehuman history. And if a person is confronted with a choice and doesn’t have knowledge that would let them do a conscious analysis (or time to do one) then gut feeling is the way to go. The reason we deride gut feeling is because people often got with their gut instead of well developed theory. That’s trusting the less-reliable source.
But if babies twitching fingers in front of their face is science, then your gut feeling is definitely science.
My point about astronomy was that we cannot do trial and error and the timeframes involved (when you go beyond our own solar system) make a lot of the phenomena that we know happen impossible to actually observe.
My main point is, is the distinction between mathematics and physical sciences really so important? Whether the rules stem from us or from the universe seems insignificant compared to whether or not there are rules that we can refine our understanding of.
I just want to jump in here real quick and point out that many mathematicians do not believe this. The question of whether math is “real” or “arbitrary” is very much not a settled issue.
I don’t think that’s accurate. Mathematics on its own can’t tell you anything is true about the real world, but it certainly can tell you certain things couldn’t be true about the real world. Because the point of math is that you get to pick whatever rules you like, but not the consequences of the rules, including whether they are consistent or not.
Personally I’m not really sold that knowledge has to be about the real world, and discovering things about cool abstract rule sets counts as learning something too.
Certainly on some level this goes back to experimental evidence…we trust the rules of logic because we have a lot of experience to show they are consistent, but have no other framework to prove that. Still, calling what pure mathematicians do science would be an example of broadening the term beyond how anyone uses it. Honestly, I don’t hear people call history a science a lot either, outside discussions about how science is the only way of knowing. I get it, you want evidence-based reasoning rather than faith-based…but is this really the way to describe that?
That’s just because, sadly, mysticism can have a hold even on people who ought to know better. But the key problem with idea of a Platonic world of numbers is that it doesn’t explain how mathematicians can create an infinite number of axiomic systems. If there was some “real” mathematics out there there could only be one true one.
I really want to know what the science behind this statement is, because that seems like a tremendous leap of faith to me. Right now we aren’t even sure whether our physical universe is singular.
I am sorry, but that is not really a response. You were presented with the statement that not all mathematicians would agree that “math itself is just arbitrary rules like chess” and you were given Platonism as an example of that. Your response was not to the underlying idea, but to the example: Platonism.
Your point seems to be that math is not the same as science because it can not teach us anything new about the world, but you have not explained this claim and, when presented with examples that suggest otherwise, you have slipped into tangents, such as who invented probability theory and the history of astronomy.
Can we stick with the point of what, if anything, separates math and science?
It means that it’s an argument that’s always true having no inverse or untrue condition by which someone can prove or disprove logically or empirically.
That sounds like an ad hominem argument…
That very well could be the case, though I’ve seen some strong counterarguments. However, I’m not here to argue for or against certain beliefs of mathematicians, just to point out that the debates within philosophy of mathematics are not in the least bit settled, and that your perspective on mathematics is not the only one. Personally, that there is such a plurality of perspectives on the issue (Platonism is not the only one at odds with your view) prevents me from having confidence in any one: I’m skeptical about our ability to settle the debate, and I believe the multitude of opinions is a result of that inability.
Anyway, it’s also worth pointing out that the debates happening in this thread have all occurred within the fields of epistemology and philosophy of science, and are far from settled as well. These are certainly engaging topics to argue about, but there is an undeniable diversity of opinions held by many very smart people.
Yes, I am trying to understand why you think that is bad, which is why I asked for an example. A statement like “2 + 2 = 4” is tautological but surely you wouldn’t begrudge me believing it or using it in an argument.
I think that a tautology is more like “2=2” or “bad fish is bad.” It is undeniably true, but also not useful.
2 + 2 = 4 isn’t tautological as you can recompose the terms like x + 2 = 4 and then through division that 2 + 2 = 4 (ex. x + 2 = 4, x + 2 - 2 = 4 - 2, x = 2, 2 = 2, 2 + 2 = 2 + 2, 2 + 2 = 4, 4 = 4). Also, simpler you can substract the twos away from each end of the equation until you resulted in 0 = 0 which also gives you a consistent identity. Simply put, you’d be proving the use of addition by other operators that are not like addition (subtraction and substitution).
So give me an example of something you consider a tautology.
(P → Q) v ( Q → P) is one such example where the argument from this proposition is always true.
The problem with it is that you gain nothing from it. There’s no way to test against it empirically if it were formed in such a fashion. I’m not sure why you’re fighting over what’s commonly understood in terms of philosophy of science and over all epistemology and logic. The fact you have a degree in an adjacent field to mine (I have a BA in CompSci) and arguing that some how tautologies are useful or necessary for scientific enterprises seems like trolling to me. They’re useful in logic but not useful in something in the realm of the empirical where the only way to know something does happen the way it happens is through the ability to be wrong (null hypothesis). When taking that fact into account you can’t take something the presumption of informal tautologies or formal tautologies (math/logic/etc) as a sound basis for a scientific argument.
That sounds like you are limiting the idea of “tautology” to argument forms rather than to arguments. That’ not the way I’ve experienced it being used. I’d call an argument tautological if it is of a tautological form.
(P → Q) v ( Q → P) is a tautological form, so (rabbits eat lettuce → it’s Tuesday) v ( it’s Tuesday → rabbits eat lettuce) is a tautological argument. I really think this is a standard use.
2 + 2 = 4, so the statement “2 + 2 = 4”, by transitivity of =, is of the form A = A (a tautological form) and is therefore a tautological “argument”.
I don’t think the word “tautology” is useful in math. We don’t make arguments in math.
I’m literally trying to understand your previous statement that somehow associates scientism with being “tautological” and that being a reason to distrust it. I’m trying to understand what you are saying. What is an example of someone who is engaging in scientism being “tautological”? My field is linguistics and when I encounter someone using a word in a way that I don’t understand, I want to figure out what they are trying to say.
I suspect that if I understood what you are saying about scientism, I would agree with it. That’s why I’m trying to understand. I’m not arguing against you.
Also, “(P → Q) v ( Q → P)” is provable so how is it different from “2+2=4” or Fermat’s Last Theorem?
