Since I promised SentientMeat I wouldn’t hijack his thread, I’m replying in this thread.
My question is: how can you claim to be analyzing the nature of existence, memories, awareness, etc and yet totally misunderstand a simple statement I made?
Show me where I said that science proves that 1+1=2.
I only said that 1+1=2, within mathematics, does not need proof. It is true based on the definitions of ‘1’, ‘+’, ‘=’ and ‘2’.
As I mentioned above, if I state
The result of the operation T(x) is called its “transform”
I define the “tranform” of 1 to be 34
Theorem: T(1) = 34
Do you seriously believe that the above Theorem needs proof?
If not, just substitute T(x) with ‘adding 1 to x’ and “transform” to “successor” and you have the same thing you had before.
SentientMeat, I’m disappointed in you. If I never claimed that you can prove 1+1=2 through science, how can you have “divested me of my mistake”?
Successor is never defined in the proof. It is what’s called an “undefined term”. Okay, one last attempt — let’s step away from this particular proof a moment: do you believe that the Pythagorean is true? If so, do you believe that it is “defined” into being true? If so, goodbye. If not, do you believe that science can prove that it will work the next time you use it? If so, goodbye. If not, then you agree with what I have said above, and we can be done with this.
You never answered my T(x) as “transform” analogy.
In that example, “transform” is never defined.
Yet, I simply define 34 to be the “transform” of 1.
Just as the link you provided simply defines 2 to be the “successor” of 1 (look at your link again you may have missed this)
What difference is there between T(x) as “transform” and “x+1” as “successor”?
Why are you picking one specific example? There are countless Theorems that are true and have very complicated proofs. They are not “defined” into being true.
But, the 1+1=2 needs no proof, at least based on the definitions in the link you provide.
Let me repeat the example again, maybe this time you’ll address it:
The result of the operation T(x) is called its “transform”
I define the “tranform” of 1 to be 34
Theorem: T(1) = 34
Do you believe that the above Theorem needs proof?
Did I mention science as proof of mathematical concepts? Why are you insisting on this strawman?
I’m not sure why you feel the need to announce your departure from any discussion. You’re announcing it here, and you announced it in SentientMeat’s thread, and you even announced your departure from the Pit.
Why can’t you simply stop arguing with someone and leave the thread without making grand statements about your departure?
Or even stay in the thread and argue with the other posters. No one is forcing you to argue with every single person posting in a particular thread. You “had” to leave SentientMeat’s because one poster was making irrational, in your opinion, posts? Couldn’t you just continue debating with the remaining posters?
The actual modern axioms of the natural numbers are a bit different from what’s been presented so far. The standard representation is something like this:
There is a natural number 0 which is not the successor of any number.
A natural number’s successor is also a natural number.
Two natural numbers are equal iff their successors are equal.
If a set S of numbers contains zero and also the successor of every number in S, then every number is in S.
And you also need the definition of addition. Let x’ denote the successor of x. Addition is defined as the operator + such that x + 0 = x, and x + y’ = (x + y)’.
Now, given that–and that’s all that you’re given–is it trivial to prove that 0’ + 0’ = 0’’? I think not, and if you try it, you’ll come to agree with me.
Note that Russell and Whitehead’s proof also required definitions of 0 and =. That’s a very modern approach, and probably overkill for this thread.
Because — again — successor is undefined. Peano’s axioms use the term without ever defining it. There are lots of theorems and definitions that may be derived from Peano’s axioms, including both the definition of addition and the conclusion that 1 + 1 = 2. But they are NOT defined in the axioms themselves. If they were, then arithmetic would not be sound. Circulus in demonstrando is a logical fallacy.
To find out whether you hold that all deduction is summarily defined into truth, or whether you held it only against Russell and Whitehead.
As an analogy to Peano’s axioms, that is a travesty. That’s my comment. What I would like to know is why on earth you believe that you instantly found something that had eluded Russell and Whitehead after considerable research, and has eluded countless others since? I mean, I’m not saying that their sheer numbers and authority prove you wrong, but isn’t it at the very least, I don’t know, a clue?
I just want to know where you stand. How am I to know what you believe about anything at all if you believe that Peano defined 1 + 1 = 2 into truth?
Because of the unscrupulousness of claiming a victory by leaving a false proposition unanswered, as you did with your transition theorem business. Even as you ask this question, you suggest that I am beholden to address that nonsense or else it must be right. It is only in that sort of circumstance that I announce I am leaving so that it be known that my exit does not constitute an endorsement of the slime that’s left behind after all the truth has evaporated.
Strike that one bit–the proof is trivial, but it’s not a matter of definition. I’m in a hurry, so I’m not writing incredibly well.
The problem with CalMeacham’s approach is that an addition table for all of N isn’t finitely expressible except as I’ve outlined above. Having what amounts to a random (in the technical sense) set of axioms is a Very Bad Thing.
Polerius, I owe you an apology. It was not you, but Orbifold who stated it so explicitly. Sorry about that, and once again, I’m grateful for fulfilling my request.
I think we have a miscommunication issue here. That example was not an analogy to Peano’s axioms but an analogy of the two definitions on the website you cited.
Let me quote the definitions:
So, even if you exclude Peano’s axioms, P1 through P5, the above definitions give us
A set called N
A symbol denoted by ‘1’, which belongs to N
When x belongs to N, then ‘x + 1’ is defined as the “successor” of x (note that it doesn’t matter if the “successor” is in N)
The symbol ‘2’ is defined as the “successor” to 1. (again, for the purposes of this example, it doesn’t matter if ‘2’ is in N)
Let’s forget we are talking about natural numbers. Just think of the above statements in terms of symbols ‘1’,‘2’,’+’.
From (4), it is by definition that ‘2’ is equal to ‘1 + 1’.
That is precisely what statement is saying. The whole essence of (4) is that the symbol ‘2’ is defined to be the result of the operation ‘1 + 1’. No proof is needed.
Misinterpretation is the word, and it is all on your side:
"Def: 2 := 1’ " means literally ‘2’ is a short hand symbol for the succesor of ‘1’, it does NOT mean " 2 := 1+1 ", which would literally mean ‘2’ is a short hand symbol for " 1 + 1 ". (Note ‘1’ is by P1-5 the symbol that represents the only element, that is not successor to anything)
However, the theorem " 1 + 1 = 2 " literally means "the addition of ‘1’ to ‘1’ is equal to the successor of ‘1’.
The definition of addition has only two rules:
A1: a + 1 = a’
A2: a + c’ = (a + c)’
Now proving the theorem requires you to use definition A1 and replace ‘a’ by ‘1’. Only this definition AND the replacement together show that your theorem is true, and therefore proves it. Hence ‘1 + 1 = 2’ is not defined, but deduced.
I am unable to clarify this any further, if you do not get it now, you probably never will. It is so obvious.
When x belongs to N, then T(x) is defined as the “JabbaTheHut” of x.
The symbol ‘&’ is defined as the “JabbaTheHut” of %.
What could (4) possibly mean by " ‘&’ is the “JabbaTheHut” of %" if not in direct context of (3), which defines “JabbaTheHut” of x as T(x)?
If the “JabbaTheHut” characterization in (4) is not the same as in (3), then we have no theorem. If the “JabbaTheHut” characterization in (4) is the same as in (3), then we do have a Theorem, but it is trivially obvious. No proof is necessary.
I am unable to clarify this any further, if you do not get it now, you probably never will. It is so obvious.
SentientMeat, if I may make a stylistic suggestion (and I hope you take this as a friendly suggestion, which is the spirit in which it is given): you use bold font somewhat excessively. I’m often guilty of over-emphasizing in posts myself, so I’m not in the greatest position to throw stones, but I think you could stand to rely on the bold a little less.
Anyway.
The same way you would falsify any other physical hypothesis. By observing that putting one object together with one object doesn’t produce three objects.
Gravity: we begin with an observation, namely that objects seem to be subject to a mutually attractive force. Furthermore, some objects can be observed to be subject to a greater attractive force than others. From this we deduce the existence of a physical quantity – mass – and define that quantity in terms of physical observations that can in principle be repeated by anyone who wishes to understand the definition. We also deduce a physical law, the law of gravitation, which predicts the behaviour of massive objects over time. This law can be confirmed (in the scientific sense, i.e. supporting evidence can be collected) or falsified by observation.
We can do the same with arithmetic. CAVEAT: I am not claiming this is how the truth or falsehood of arithmetic is dealt with in modern mathematics, because it’s not. But the following is possible and scientifically perfectly fine:
Arithmetic: we begin with an observation, namely that collections of objects have the property of being “numerous”. Furthermore, some collections of objects can be observed to have the same type of “numerous-ness” as others: in modern language and in the spirit of Cantor, some collections of objects can be observed to be in bijective correspondence. From this we deduce the existence of a physical quantity – cardinality – and define that quantity in terms of physical observations that can in principle be repeated by anyone who wishes to understand the definition. (Namely, putting sets in bijective correspondence with one another, as Cantor did. There’s no set of platinum-iridium slugs in Paris solely to define the number “three”, but there’s no reason why there couldn’t be if we chose to define mathematics scientifically instead of logically.) We can also deduce the “physical law” that “1+1=2”, which in this context is a prediction about the behaviour of collections of objects. Namely, if I combine two collections with the observed cardinality of “one”, I’ll get a collection whose observed cardinality is “two”.
I’m defining two and three, yes. But I’m defining them based on physical observation, just as I define mass based on physical observation. And I can deduce the laws of addition from the behaviour of collections of objects, just as I deduce the attraction between massive objects from their behaviour.
I’m not claiming that “2” can be proved scientifically. I’m claiming that “1+1=2” can be proved scientifically.
But that’s a key point. How do we prove the existence of “mass”, scientifically, in a way that can’t be equally applied to prove the existence of “cardinality”? Both are, yes, definitions which are based upon physical observation, namely that some objects behave similarly (be in bijective correspondence/be subject to the same amount of attractive force) in similar circumstances and some do not.
Besides, it’s explicity the law of gravity to which I’m comparing “1+1=2”, not mass.
No falsifiable consequence? Stick three protons and two anti-protons in a particle accelerator and see if any particles are left over afterwards. Now do the same with two protons and two anti-protons. There are definitely physical, observable consequences which distinguish between the two cardinalities. Better yet, tell me there’s no difference between uranium-235 and uranium-238. The difference is only a matter of the number of neutrons in the nucleus of the atom. If there are no observable, falsifiable consequences to the difference between “235” and “238”, then I’m sure there will be no difference in the behaviour of each isotope.
Saying it’s only a matter of asserting a definition is like saying that arguing over gravity is only a matter of asserting the definition of “attraction”. I can redefine “attraction” to mean “blowing candy kisses while singing a duet” and then claim that massive objects don’t demonstrate “attraction”. Doesn’t mean the existence of gravity can’t be scientifically proven.
I’m busy as spit today, so I’m not going to be able to reply much for a while. In particular, eagle’s charming “piece of crap” comment will have to wait.
What’s with all the deducing? I thought you were going to use science.
Just so you know, deduction is in the purview of logic, not science.
How? What test will show that if you add 1 + 1 after the test has been completed, you will still get 2? In other words, how does your test prove that you’ll get 2 the next time you try?
Science can’t prove the existence of anything. Existence is an ontological claim.
You are mixing up what is being defined with what it is defined to be.
T(x) := JabbaTheHut of x
& := JabbaTheHut of %
Theorem: T(%) = &
Proof: use 3) with ‘x’ replaced by ‘%’ which gives T(%) = JabbaTheHut of %, therefore we can replace ‘T(x)’ in the theorem with ‘JabbaTheHut of %’.
Hence, the new deduced theorem, to prove: JabbaTheHut of % = &
This does not need further proof, because it is 4) with sides just switched.
So still, even with your phantasy definitions, you need a single step, where you use a definition/axiom and a replacement, to finally convert the theorem into another definition/axiom. Proof is necessary.
I have to go now.
cu
ps: Try to learn Prolog. This programming language is very helpful to understand predicate logic and therefore how to prove theorems from axioms/definitions or facts.
Ah, poor Polerius. You asked a simple and, I think, very good question:
and people got sidetracked and hijacked and focused on the specific example you gave (1 + 1 = 2) rather than the underlying question itself.
I guess it’s related to (or at least the same sort of question as) the question of what constitutes a valid mathematical proof—how rigorous do you have to be in justifying each step? And this is far from easy to answer.
Indeed. Russell and Whitehead took (I believe it was) 362 pages of meticulous inferences to prove that 1 + 1 = 2, only to have their entire system smashed to smithereens by Godel.
No, that’s like saying that in France it equals deux, and that therefore something is amiss. The Peano axioms apply to any base that can accomodate the natural numbers as defined. 10 in base 2 is the same value as 2 in base 10.
eagle and Lib have it exactly right. What you are doing is overloading “+”. Fundamentally, 2 = succ(1) is not the same as 2 = 1 + 1. The successor function is an axiom, as Lib says. Addition is defined in terms of the successor function. I think you see that 4 is not defined as being 2 +2, but as succ(succ(succ(1))). Since you interpret succ(1) as being the same as + 1, you get confused as to the distinction between what is axiomatic and what can be defined from the axioms.
Think of it this way. If you wrote a little theorem proving program to prove that an z = x + y based on Peano, you can throw in any three numbers and either come out with the result that the statement is false or the proof. You would throw in 2 = 1 + 1 and come up with the same class of proof, though it would be trivial. You would not be able to do any such thing with 2 = succ(1), since that is axiomatic.
I hope this different view helps everyone see the difference.