John responds to me:
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And?
And you've missed the point is several different ways.
Strange...I was going to say you had missed the point. But since you seem to be arguing my interjection rather than my explanation, I am at a loss for how to respond.
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How does this negate the fact that by being a square, it is necessarily not a circle?
This is true only within a defined ruleset-- a defined ruleset which, by the way, is itself self-contradictory.
Says who?
You're not about to invoke Godel, are you? Be careful, because the Incompleteness Theorems do not say that all axiomatics systems are necessarily incomplete or inconsistent.
Instead, they say that all axiomatic systems sophisticated enough to model arithmetic are such. Not all axiomatic systems are that sophisticated and, indeed, they are both complete and consistent.
When did we agree on the axiomatic system?
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There is no guarantee that it applies to the real world. Sorry. Your incredulity is irrelevant.
I was going to say the same thing to you.
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The second way you missed it, and more directly the intent of my statements, is that the functional component of a proof-- any proof-- is the proving of something about the relationships between your premises-- proving a true statement about that/those relationships.
I know. I have always said so. You can't prove anything without a premise. Are you about to embrace Cartesian Doubt?
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Any proof, if it is correct, is a true statement-- true not false, positive, not negative.
But by being true, it necessarily results in certain other things being false. Proof draws a boundary. Thus, there is a distinction.
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Constructing a proof, by default means constructing a true statement.
And that statement just might be "It is true that this is false."
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One can construct a hundred false arguments
But we're not talking about all of those. We're only talking about this specific one. If we're having a discussion about the color of my hair, it doesn't matter how many statements, true or false, you can make about my sister's hair. We're not talking about my sister's hair. We're talking about mine.
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-- the conclusion does not follow from the premises-- and it simply does not matter. The arguments are irrelevant. They don't matter. The only ones that do matter are the true ones.
But sometimes the only way to show something to be true is by showing something else to be false.
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You have to keep your implications going in the correct direction.
And you have to think more carefully about the statements being made.
Strange. I was going to say the same thing to you. Now that we have the ad hominem comments out of the way, can we get back to the matter at hand?
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If I say 'square' you know what I mean. You can draw it. If I say 'not a circle' it conveys virtually no information.
No, it does convey some information. And if what we're looking for is circles, then we don't need to know any more.
If I'm trying to show you that 2 + 2 != 5, it is not necessary for me to show that 2 + 2 = 4. While that would, indeed, be sufficient, it is not necessary.
You do understand the difference between sufficient and necessary, yes?
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It eliminates one of an infinite set of shapes.
Sometimes that's all you need.
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They are quite different statements.
I know. But you're arguing a completely different point.
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If you can draw 'not a circle' in the same way that you can draw 'square' -- with a compass and strait edge-- then lets see it. Until then, you cannot claim they are the same type of statement.
Who said anything about drawing anything? You're arguing something completely different. Please come back to the subject at hand. I have no idea where you're going.
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Who said anything about the subatomic level?
If you want an adequate definition, you need to include all components in the definition.
Incorrect. You merely need to include all the necessary components.
You do understand the difference between necessary and sufficient, yes?
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Incorrect. If the statement uses the existential operator, then it is an existential statement. That is by definition.
You are not using a relevant definition
Excuse me? We're talking about existential mathematical statements and somehow the existential operator isn't relevant?
Did you really just say that?
If you cannot understand how an existential mathematical statement is one that uses the existential operator, then I really don't know how we can continue.
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and are stubbornly refusing to use, or to even attempt to understand, the relevant definition.
Strange...I was going to say the same thing to you.
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There is no largest prime number. It does not exist.
What you have proven is that if there is a largest prime our system of mathematics is contradictory.
Indeed. It is written as a negative existential statement:
There does not exist
x such that there is no prime number larger than
x.
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You assume a largest prime and prove a contradiction, and set this up so that it applies to any prime you fill into the blank. What you've actually got is an infinite series of assumptions and an infinite series of proofs that there is a larger prime than the assumed one, or for any prime there must be a larger one.
Incorrect. As my Fundamental Concepts professor so often pointed out to me, a proof with an infinite number of steps isn't a proof. We do not even attempt to "fill in the blank" with a specific number because the specific number is not necessary. In fact, to give a specific number will defeat the purpose. Sure, you may have proven it for that specific number, but what about this other specific number? Until you manage to do so for all numbers, of which there is an infinite number, you haven't shown what you are trying to show.
So you don't be specific. It doesn't matter what the number actually is. It is not necessary to know. The only thing that is necessary is that it have the trait of being the largest prime.
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You can translate this into English as a negative, but that isn't how the proof works. The proof is an infinite series of positives.
Incorrect. It is a finite series of deductive statements leading to a contradiction which necessarily requires the conclusion of a negative existential statement.
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So? One can show that an infinite number of things are of a certain set by showing that they are not members of the complement.
So... do this in the real world.
We do this all the time.
Take the set of me. There is me and the complement of me, "not-me." What are the elements of "not-me"? They are the infinite possible things that are not in the set of "me."
Besides, I gave you some real world examples...or do you claim that the objects of mathematics aren't real?
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As Holmes put it, when you have eliminated the impossible, whatever remains, however improbable, must be the truth.
Holmes was not really much of a logician.
But he's right. And in fact, that's precisely how science works: Science isn't about showing things to be true. It's about showing things to be false, leaving only the truth behind. Since science works as an observational, inductive method, it can never be sure that it has actually arrived at the truth. Just because observation is consistent with prediction doesn't mean the theory is "true." So instead, science sets out to chip away the false things so that what is left behind is the most consistent with observation.
I'm afraid I'm going to have to be doing a lot of editing of your response because I literally do not understand where you are coming from. You have hacked my post down to such small statements, including single words, eliminating all context, that I am having a hard time following you.
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I can tell you everything about what a "square-circle" is: It has all the properties of a square and a circle.
Can you? How?
Like I just did: It has all the properties of a square and a circle. What more do you need to know that isn't taken care of by the definition?
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How do you know the properties of a square?
By it's definition.
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Initially, someone measured one, then another, and generalized.
No. Do I need to crack open
Elements and give you the definition?
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Same with circles, same with triangles. And we end up with something we call geometry. It is the generalization from specific cases to the universal and as such will always be questionable. Hate to break it to you.
I hate to break it to you, but there was this guy named Euclid and he collected the works of other mathematicians, as well as coming up with some work on his own, and came up with an axiomatic system of geometry by which circles, triangles, and squares were not a "generalization from the specific to the universal" but rather were definitions. Perhaps you've heard of the book...it's the most popular book in the world after the Bible. It's called
Elements. It provides definitions of circles (definition 15), triangles (20 and 21), and squares (22).
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Without a square-circle to measure, you can't know with certainty, its properties.
And yet, I do. By definition, a "square-circle" is an object that has all the properties of a square and a circle.
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You can't know if you've named all of them, or if you've got all the properties correct.
Sure I can. I simply need to provide a definition that is necessary and sufficient. And I have: A "square-circle" is a plane object that has both the properties of a square and a circle.
Now, if you wish to backfill those definitions, here we go:
Of quadrilateral figures, a square is that which is both equilateral and right-angled. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another.
Ah, but that requires us to know what a "quadrilateral figure" is and what a "straight line" is.
Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.
But, this requires us to know what "rectilinear" is.
And when the lines containing the angle are straight, the angle is called rectilinear.
But, this requires us to know what a straight line is.
A straight line is a line which lies evenly with the points on itself.
But, this requires us to know what a line is and what a point is.
A line is breadthless length.
A point is that which has no part.
So there you go.
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You can't check your assumptions, in other words.
Of course not. That's why they're called assumptions. If they were derived from some place else, they'd be conclusions.
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What you've got is thought experiment. What you've got is "if the assumptions of plane geometry are true, then square circles can't exist." Notice how the whole structure is conditional, and that there is no guarantee that it applies to the real world?
Are you saying planes don't exist in the real world? And no, let's not be disingenuous and talk about aeroplanes. I'm talking about the flat surfaces one might find in a discussion of plane geometry.
After all, there's a whole bunch of mathematics surrounding the substitution of Euclid's Fifth Postulate with something else. But you end up with something that isn't plane geometry.
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But such an object still doesn't exist.
hmmm... if space were to be contracted to a point as it is a a singularity, would not the dimensions of a square and a circle match?
Nope. A circle is contained by a line. A line has more than one point. If space results in a single point, we have no circles, no squares, no lines. Simply a point.
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The radius of a circle under such circumstances would be zero, as would any measurement you made of a square. Thus, they would appear to be the same.
No, squares have four sides, not one. Even in a singularity, we can't have a "square-circle" because you can't have a square. And since "square-circles" need to have the properties of squares, then if squares can't exist, neither can "square-circles."
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Information only exists for what actually is defined.
Why are you equating definition with existence?
I'm not. You're reversing the direction again.
All squares are rectangles, but not all rectangles are squares. Things that exist have definition, but not all things that have definition exist.
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No, what is meant by "you cannot prove a negative" is acceding to the logical error of ad hoc argumentation.
It is getting damned frustrating to attemp a debate with you when you choose to remain ignorant of the meaning of the phrase which started the debate. You can't win by arguing against the wrong definition.
Strange...I was going to say the exact same thing to you. You keep refusing to deal with the question at hand but instead want to deal with all of these side issues, throwing ad hoc comments in as if they had any relevance.
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You mean Descartes was wrong?
Frequently.
See, this is what I mean about you hacking my post to shreds, completely devoid of context.
Let's try it again:
You mean Cartesian Doubt is logically valid?
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A difference that makes no difference really is a difference?
Could be.
How? There is no difference.
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Of two things which appear to us to be the same, or make no difference, one or the other may well be true and the other false.
Ad hoc. This argument is akin to the one I'm having with crashfrog where I am asking him if we have one apple and add one apple, do we not get two apples. His response? It isn't an apple! It's an orange!
Excuse me? Since when did it become an orange? We weren't talking about oranges. We were talking about apples. Indeed, if we take one apple and and one orange, we don't get two apples (though we do get two fruits). But once again, we're not talking about oranges. We're talking about apples. If what I added was an orange, then the scenario of "take one apple and add one apple" is not satisfied and we should not be surprised to find that we've come up with a different result.
If I take one apple and add one apple, is it possible for me to get something other than two apples?
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There are cases where lack of evidence for S is relevant to the truth or falsity of S.
What would those cases be?
I gave you examples. That you refused to read them is not my problem.
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Or that I wasn't proving the non-existence of something?
... that you are proving only within a defined rule set. And that you are translating positive mathematics into negative English and calling it proof of a negative.
But I can state it negatively in mathematics, too:
Given a factorization
P of an integer
n, there does not exist
x such that
x > max(
p element
P) for all
n element I.
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But if the thing's existence requires that we have evidence and we don't, then we necessarily conclude that it doesn't exist.
More like 'assume.' Unless you have absolute and infallible knowledge you are arguing from ignorance.
And why is it impossible to have absolute and infallible knowledge?
Is that not what definitions are for?
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It is tautological, but it is not pointless. It is not trivial.
Sorry, but it is.
Then why do I have so many textbooks that ask you to prove that A = A? Surely they think there is a point to it.
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That is because that's precisely what happens. By proving what something is, you necessarily prove what it isn't.
Then try doing it the other way around.
No, let's not.
By turning around, you risk running into a different result. All squares are rectangles, but not all rectangles are squares. Sometimes it's easier to define something by what it is. Sometimes it's easier to define something by what it isn't. Take, for example, the example I gave you about finite and infinite sets. It's easier to define what a finite set is by describing what it is: A set whose elements can be mapped to
Jn for some
n. However, it's easier to define what an infinite set is by describing what it is not: A set that is not finite.
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How is that insufficient?
Mathematics is a made up system.
Once again, you have hacked my post to shreds so small that all context has been lost.
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It is based upon assumption and is internally inconsistent. How exactly can it be SUFFICIENT?
Because you apparently do not understand the Incompleteness Theorems.
They do not say that all axiomatic systems are necessarily inconsistent. Rather, they say that only those that are sophisticated enough to model arithmetic are.
Not all axiomatic systems are that sophisticated and, sure enough, they are complete and consistent.
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I am arguing the validity of deduction and how one can logically deduce negative propositions.
Then you are not arguing the fundamental point.
Yes, I am.
The claim was a universal. It was not that you can't induct a negative. It was that you can't even deduct one.
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Incorrect. Even at singularities, logical things happen.
You have got to be joking?
Nope. Even randomness and chaos behaves logically.
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Question: Do you know about the mathematical concept of "Platonism"?
Question: Do you know that Platonism isn't a mathematical concept?
Indeed, it is a philosophical concept, but it is a common discussion among mathematical circles. Please forgive my sloppy phrasing. That said, could you answer the question:
Do you know about Platonism and how it relates to mathematics?
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A Platonist would then say that there is an answer to the question, we just don't know what it is.
This is just about the most pitiful thumbnail version of Platonism I have ever seen.
So? I'm not saying it's the most elegant example out there. I'm simply providing it as a way to describe the difference.
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The size of the Reals does exist, we just have no tools to let us know what it is.
So... there is a largest Real number then?
Excuse me? How does one get from the size of a set to the value of the elements within the set?
You do understand that the bag you carry your apples home in is not the same as the apples it contains, yes?
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Thus, inside the event horizon of a black hole, at atomic scales, etc., things happen according to their internal logic...we just don't know what it is.
And you know this how?
Because there is no other way for them to behave. Are you saying they behave according to the rules of external logic? That we can think our way into forcing the behaviour of a black hole to conform to those thoughts?
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Because you say so? Because you think so?
No, because logic says so. Black holes exist. Therefore, there is a behaviour on black holes. Therefore, there is an internal logic to that behaviour.
Even randomness and chaos behave in a logical manner.
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I don't know if there is an internal logic or not, but that logic, if it exists, is damn sure not the logic we know and love.
Sure it is. What else could it be?
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I'm not sure what to call this... equivocation? Arguing from ignorance? Just plain 'it is cause I say it is'?
I've got a better one:
You not understanding.
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Every time you insist that something is mathematically or logically impossible and thus it IS IN FACT impossible, you are implying that math and/or logic is in fact an accurate representation of the world.
Is there a way it couldn't be?
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You seem to be arguing that because I cannot disprove the existence of any possible description of god you might be able to come up with, that means I am incapable of disproving the existence of any specific description of god.
This is the root of the most common problem with attempting to prove a negative and probably the impetus for most chat-room utterances of the phrase. You do end up being asked to disprove an infinite series of 'nots.'
So you
are saying that because I can't disprove an infinite number of things, I am incapable of disproving a finite number of things.
That seems to be the problem. You're looking for a disproof of all possible things while I'm simply referring to the specific one at hand. You don't have to do this "infinite series" of yours because we're not dealing with an infinite number of objects. It's only one.
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I don't hve to "observe and record everything that is, was, and will be across all of space and any other spaces there might be" in order to define a square.
But to make some claim of actual existence or non-existence, you do.
No, I don't. I simply need to define the extent of my disproof. For example, the proof that my car keys are not in this room is only good for this room. It may or may not apply to other rooms.
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Otherwise what you have is a generaliation from the specific to the universal.
Who's jumping from the specific to the universal? I'm certainly not.
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I can't be more plain than that. This is induction's dirty little secret.
I'm not talking about induction, though. I'mm talking about deduction.
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Yes.
Numbers exist? Show me one.
Assuming you have a typical hand, take a look at it.
You'll see five.
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By the way, "inconsistent" is equivalent to "incomplete." It all depends upon how you look at it.
Ya sure about this?
In the sense of the incompleteness theorems, yes. You can choose which way you want to go. You can write it out such that you get A and ~A or you can write it out such that you cannot say either A or ~A. They're the same thing.
It's kinda like various statements of Euclid's Fifth Postulate. You can state it in a whole bunch of ways, but they all amount to the same thing. Non-Euclidean geometry is, primarily, about actually changing the substance of the Fifth Postulate.
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Rrhain
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