Well, ok, just in order come to an agreement between us, let’s modify this a little:
IIRC it was Galileo Galilei who formulated the formula how velocities are added: vResult = vOne + vTwo.
This formula had been “proved” scientifically to be useful. It was actually that useful, that you could have come to the conclusion, that it is correct.
Galilei also designed a test for the speed of light, but it took 250 years until Fizeau and after him Focault actually were able to measure the speed of light. And it took 30 more years until Michelson showed that light always travels with the same speed (actually his test was improved and repeated seven years later joining with Morley).
This experiment was very much debated, because it contradicted the “truth” behind the above formula.
It was finally Mach who was the first one to interpret the results of the Michelson/Morley experiment correctly. He replaced the Galilei transformation with the Lorentz transformation and so he got the famous relativistic formula how to add velocities, but he did not realize the all the consequences of doing so.
It was then Einstein (who did not know about the Michelson/Morley experiment) who merged the Lorentz transformation into a complete theory of relativistic mechanics. He realized, that masses grow with relative velocity, and that synchronity of events does not exist. This was his famous special relatvity (SR).
Finally with SR we now had a formula, that did no longer contradict the Michelson/Morley experiment and that (for small velocities) did differ from the “proven” Galilei formula only so little, that these differences were not measurable.
So, if you say to “scientifically prove” a formula means, that it does not contradict experience, but does not mean that the formula is actually correct, then we agree.
A mathematical proof, however, is an exact proof. If you can prove something mathematically, then it is true and correct and final. Nothing will ever be invented or experienced, that makes this proof false.
To me, scientifical proof, is not a proof at all. It is a deduction from experience, that can induce (i.e. predict) results of experiences to make in the future. But I see, that you can have the opinion, that this qualifies to be called “proof”, only I do not share this opinion.
I know that’s what you mean. Let me give you, for argument’s sake, that you have indeed, by your scientific test, proved that 1 + 1 = 2. Now you must prove that 1 + 2 = 3, that 1 + 3 = 4, that 1 + 4 = 5, and so on until you’ve proven that every natural number, N, plus every other natural number, N’, is equal to N + N’. Once you’re done with that, you need to conduct similar tests for the integers. And then for the rationals. And then for the reals. And so on. Got time?
But equations (and logics in general) allow for that simply by changing the axioms and/or rules of inference. Taking your example of a circle, non-Euclidean geometries describe circles on spheres and saddles and other topologies in addition to flat planes. Parallel lines are postulated differently. Rules of inference vary among systems as well. Some systems, for example, do not allow you to invoke the excluded middle. In such systems, A OR NOT A does not hold — there might be some truth value to A other than true or false (like partially true, for example). And there are modal logics as well, built on the Kripke system by adding postulates with respect to modalities, so that we can model things like necessity, obligation, knowledge, and belief. There’s a lot more to logic than simple 2-value propositions.
A better word is “induce”. What you do with scientific tests is induction. You do some experiments that evoke a pattern so that, for example, you eventually test enough natural numbers that you draw the conclusion that you’ve hit upon a generality. You conduct a thousand tests and reason that what you’ve done must apply more broadly.
Induction reasons from the specific to the general. Deduction goes the other way, from the general to the specific. Deduction goes like this: all squirrels are rodents; Rocky is a squirrel; therefore, Rocky is a rodent. It proves that Rocky is a rodent on account of rodents being more general than squirrels. Induction, on the other hand, goes like this: I’ve examined an awful lot of squirrels; every single one has turned out to be a rodent; therefore, Rocky the squirrel is quite likely a rodent. That’s why deduction is called proof, but induction is mere evidence.
Now. That said. Deductive systems are built on top of induction anyway because, as you’ve seen, deduction has to have something to deduce from. That’s what the axioms are for. They serve as a place to start. They are propositions that the proof builder decides will be generally accepted as true. Axioms are offered without proof. They are deduced from nothing. They are statements that the proof builder hopes ring true inductively. It is in this area that science can be helpful — testing the axioms for reasonableness. (But not truth.)
Take Einstein’s Special Theory of Relativity. It begins with two axioms: (1) physical law is the same for all reference frames, and (2) the speed of light is constant in a vacuum. It was by beginning from these two axioms (also called postulates or premises) that Einstein deduced the relativity of simultanaeity.Other inferences follow as well, including the spectacular e=mc[sup]2[/sup]. He did all this with a pencil and some paper. No experiements other than in his head. He used deduction (in the form of both logic and mathematics). Here is an excellent overview of both his theories, including explanations of how they are deductively derived and then tested inductively. Science can test whether the theories are false, but only deduction says whether they are true.
Finally, take the famous proof that is the topic of this thread. One obvious question that arises (though it hasn’t arisen here) is, how does the proof of 1 + 1 = 2 show N + N’ generally for all natural numbers? And the answer is, it is in axiom number five — possibly the most well known and important axiom in all of mathetmatics: The Induction Axiom. In plain English, it states that every natural number will always have a successor. This axiom, like all others, is offered without proof. And with respect to it, your scientific tests would be useful. But not conclusive!
How do you know? What consequences would be observable if it did? Or are you saying it’s impossible (in which case, yet again, you are imposing 1+1 != 3 onto reality)?
Stop a moment. Are you saying we deduced that massive objects attract? That it was previously considered possible for massive objects not to attract? And we came to agree that they do by observation, rather than simply declaring it to be impossible for massive objects not to attract?
And how do we decide who has counted correctly, again?
But we didn’t disagree on the definition of “attraction”. Before we ever made observations, we would have had to admit that massive objects might not attract each other - that that was possible. Is it possible for 1+1 to equal 3 before any observations?
Nope. In fact, I’m pretty sure I already said to eagle something to the effect of “If I insist on a scientific viewpoint of mathematics, then that will put horrible limitations on the mathematics I can do.” This is, however, irrelevant to my claim that “1+1=2” can be approached from a scientific perspective as well as a mathematical one.
You know, when I see these two paragraphs together – me, explaining that “induce” would be the more exact term, that induction is part and parcel of the scientific method, but giving my reasons for preferring “infer” over “induce”, followed by you, telling me that “induce” would be the better term, explaining that induction is a key part of the scientific method as if I hadn’t just said that in the previous paragraph, and completely not addressing my stated reasons for not using the word (followed by a lecture on induction vs. deduction that leads up to a triumphant presentation of the principle of mathematical induction as if I haven’t taught the principle of mathematical induction in a classroom setting, which I have…here’s a tip, I know what induction is, thank you very much) – I can’t help but wonder if you’re actually reading my posts.
In order: from a scientific viewpoint, I don’t know prior to performing the experiment. The consequences? Presumably some very weird things would happen in the world of chemistry, just for a start. And no, I’m not saying it’s impossible, from a scientific viewpoint. I have no reasonable mathematical model which doesn’t say 1+1=2, but there’s always the possibility that the model might not conform to reality.
Yes. (However, before Liberal comes running I would say “infer” rather than “deduce”.) In the time of Ptolemy, it wasn’t conventional wisdom that Mars and the moon were connected by an attractive force. That knowledge had to be inferred from observation.
Now, if there was a time when it wasn’t conventional wisdom that “1+1=2”, I’m not aware of it. However, that doesn’t change whether or not it is possible to consider the statement “1+1=2” from a scientific standpoint.
If we’re considering the question from a scientific standpoint, then either one of us decides by comparing to a common standard. Just as either one of us can decide whether or not we’ve weighed a test mass correctly, because we have a common standard to refer to.
Yes. In the context of a physical experiment and from the viewpoint of the scientific method, it is conceivable/possible/whatever that 1+1 might equal three, prior to observing the outcome of an experiment which tests “1+1=2”. Just as it is conceivable/possible/whatever that acids and bases might not neutralize each other, that masses might not attract, or that light might not refract in a prism, prior to observing the outcome of an experiment which tests one of those hypotheses.
From a mathematical or logical viewpoint, the answer is very different. And even from the scientific viewpoint, in practice no one has ever observed an outcome that didn’t confirm “1+1=2” and I would pretty damned startled if anyone ever does, while on the other hand everyone observes outcomes which do confirm “1+1=2” all the time. But before a hypothetical observer sees such an outcome, from the standpoint of science such an observer must admit the possibility that 1+1 might equal 3. Any certainty that it won’t doesn’t come from science: it comes from intuition, logic, or faith.
Argh! Listen to what you just said. Before any monkey put any object next to each other, he had an idea that 1 + 1 might be 2, or possibly 3. He puts one thing next to another and, lo and behold, discovers that it’s 2 rather than 3.
If you can’t see what absurd cart-before-horsemanship this is, I’m not sure I can put it in any more stark a manner. (Incidentally, I am a scientist - I’m not some philosophy addict trying to trip you up.)
Thanks for sticking with this, SentientMeat. I have been following this whole thing, and the twists and turns have been driving me crazy. It is to blame for why I keep contributing to the Mad Scientist thread.
Yes, but he also said that god does not play dice with the universe, and as my-reach -back-in-time-o-scope has clearly proven, he clearly plays Parcheesi.
What!? No, it is not just a VCR hooked up to the shell of a 1920’s style deathray, zoomed into a hand playing Parcheesi against a green screen background. It is surplus from The Green Lantern Corps yard sale. I just modified it. And it is not a silly name. It is a direct translation from the Maltusians. [Secret Origins Vol. 2 #23]
Ok, so maybe this is just a buncha’ comic book referenses. I am unrepentant.
Why is this “cart-before-horsemanship”? What scientific reason does the monkey have to reject the idea that 1+1 might equal 3, before he sees it for himself?
The fact that 1+1=2 follows logically from the axioms and rules of inference of mathematics isn’t a scientific reason. (The model, after all, may not be faithful to reality, just as the inverse square law isn’t completely faithful to reality.) The fact that we have a hard time imagining how it could be otherwise isn’t a scientific reason. (We had a hard time imagining how distance could be relative, prior to Einstein.) The fact that modern mathematics doesn’t use such a standard of proof, as your Einstein quote suggests, is not a scientific reason for rejecting the idea. The fact that neither of us would ever believe in a million years that the experiment could have any other result than the one we expect is not a scientific reason for rejecting the idea that 1+1 could equal 3 before performing the experiment.
I’m not accusing you of being a philosophy addict trying to trip you up…if anything, I’m the one who’s being a stickler about the philosophy of science here. Of the two of us, I’m the one who’s being a stickler about what conclusions are the domain of mathematics and logic and what conclusions are the domain of science. And concluding that 1+1 won’t equal 3 before performing an experiment to test that hypothesis is not a scientific conclusion.
It’s a perfectly good conclusion for a whole host of reasons, but it is not a scientific conclusion.
just to put something into this debate about how scientific proof of 1+1=2 could be falsified: Take one gallon of distilled water and one gallon of pure alcohol and put them into a large enough barrel. What do you get?
a) exactly two gallons of fluid? (which would prove 1+1=2)
b) less than two gallons (which would falsify 1+1=2)
c) more than two gallons (which also falsifies 1+1=2)
Think about it and you will see, scientific prove of 1+1=2 is actually impossible.
You can prove 1 pebble + 1 pebble = 2 pebbles, and a lot more things. But the natural numbers 1 and 2 do not exist in reality. They only exist together with units. Another example:
1 man + 1 woman = 2 people = possibly 1 family = or still 1 man + 1 woman.
You cannot design a scientific test for 1+1=2 that does not use any kind of units. And as soon as you have to include units into your test, you do no longer test 1+1=2 but actually 1 [unit A] + 1 [unit B] = 2 [unit C], which is very much different and not always “provable” but actually wrong.
And even if all these units are of the same kind this calculation does not need to be provable:
Definition of a unit of velocity: ReallyFastVelocity (RFV) = c/2.
1 RFV + 1 RFV = 2 RFV is not true (using SR you actually will calculate/measure: 1 RFV + 1RFV = 8/5 RFV)
Now, do you still think you can “prove” 1+1=2? If yes, please explain your test scenario, that does not involve units of any kind.
This thread reminds me of one in which one or two posters kept insisting that .99999… is not equal to 1, despite a variety of proofs, references, and explanations. This one and that one were equally maddening.
But how does laying one object next to another conclude anything? Our “massive objects attract” hypothesis is eminently testable: we get two big iron balls on long wires and use interference fringes or the like to seek a verifiable consequence.
Our 1+1=2 “hypothesis”, on the other hand: We lay one object next to another and … hey presto, we have one object next to another. One can only define one object next to another as two objects. You don’t discover it by observing the consequences.
I can assure you, I’m also feeling quite a bit of frustration that my point is clearly not getting across. Nevertheless I’m going to spit into the wind one more time:
Why is it so hard to understand that the hypothesis “when you put a set of cardinality one next to a set of cardinality one, you end up with a set of cardinality two” is testable? The conclusion, the verifiable consequence, is that you don’t get three objects.
This is not just a matter of the definition of two and three any more than gravitational attraction is just a matter of the definition of mass. Your proposed experiment with the iron weights isn’t testing the definition of mass. What’s being tested in my proposal is not the definition of two or three, but the process of addition. What’s being tested is the idea that our mathematical model of addition is an accurate model of the real-world process of combining two sets of objects. To use your words, what we discover by observing the consequences of that real-world process is that the hypothesis predicted by our mathematical model has come to pass.
Dammit, read your own Einstein quote! What we test is the degree of certainty which mathematics refers to reality! Einstein understood: the mathematical model and the real-world process are different things. The model provides scientific hypotheses concerning the behaviour of the real-world, but it’s always at least conceivable that the model may be inaccurate. And because of that, it’s always possible to test the hypotheses provided by the model by comparing them to the real world.
“F=-GMm/r^2” is a statement in a particular mathematical model of physics, but it’s a testable hypothesis in the real world (one which is only accurate to a certain degree). “C=2(pi)r” is a statement in a particular mathematical model of geometry, but it’s a testable hypothesis in the real world (and again, one which is only accurate to a certain degree). “1+1=2” is a statement in a particular mathematical model of arithmetic. Why, then, do you refuse to consider even the hypothetical possibility that the model may not be a 100% accurate representation of reality? Why does arithmetic get a free pass when geometry and dynamics don’t?
There are observable, verifiable consequences, namely that we don’t end up with three objects. There is a meaningful conclusion, namely that the mathematical model accurately predicts the results of the real-world process. So what, then, separates arithmetic from geometry and dynamics? What is the difference?
That may be what the conclusion is, but here is what the conclusion isn’t — that the NEXT time you are ASSURED of getting two again BEFORE EVEN STARTING. Every experiment tells you only what you got when you tried, not what you MUST get every time you try.
As I’ve said before, by observing the outcome and measuring it according to an agreed-upon standard.
Seriously, what is the point of this question? I perform the experiment. I observe the results. I measure the results according to an agreed-upon standard. One conceivable outcome is that the resulting set has the same cardinality as a “standard” set of two objects. Another conceivable (and I stress the word “conceivable”) outcome is that the resulting set has the same cardinality as a “standard” set of three objects. Are you suggesting that the fact that we have an agreed-upon standard definition of “two” and “three” somehow determines the result of the experiment before we perform it? If so, why? And why doesn’t the fact that we have agreed-upon definitions of “length”, “radius”, and “circumference” similarly determine the radius of a real-world circle before we ever measure it?
You seem to think I disagree with that. I do not, and I don’t see how any of my posts in this thread would make you think that I do. In fact, I’m considering asking you to repeat that statement to SentientMeat, as it would emphasize the point I’m trying to make to him. However, since I have said in this very thread both “I agree that it’s entirely possible, logically speaking, that one day gravity will reverse and the Earth will spontaneously explode”, and “Yes, a scientific hypothesis can be overturned at any time by new data”, why are you addressing this comment to me?
Finally, eagle: I’m not ignoring you, I just don’t really disagree with anything you said in your last post. The fact that cardinality in the real world tends to imply a unit, and that sometimes units are fungible, just emphasizes my point that there’s a conceptual difference between the mathematical model of “1+1=2” and the real-world phenomena we might try to model with it.
It does not, however, change the fact that “1+1=2” is both a mathematical statement (when viewed from the context of the mathematical model) and a scientific hypothesis (when viewed from the context of real-world phenomena). Which is what I’m trying to explain to SentientMeat.
OK, Orbifold, let’s settle this between us: I can agree with that, as long as you agree that scientifically prove 1+1=2 through an experiment for example with pebbles not actually proves 1+1=2 (which can only be proved mathematically) but instead it proves the scientific hypothesis, that pebbles as cardinalities follow the ‘law’ of 1+1=2 (or more general of a+b=c, which can be further verified with more experiments).
