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Hi everyone, before you read on let me warn you, this will probably be a boring blog to some of you because it is math related. i dislike math myself, even though i know how useful it can be. i just wanna discuss a couple of aspects of it with the smart TL community that cares enough to read.
Anyway. I remember having a discussion in a class regarding the topic of TRUTH. The discussion then derailed into the subjects of Mathematics + Physics and their proximity to perfection or truth, because most students argued that these sciences could prove everything. I understood their argument completely for obvious reasons, but then i asked the following:
- if math/physics is such accurate systems, why do they have such innate flaws? for example, x / 0 is undefined, etc.
- why would math/physics be any more special or accurate when they are nothing but a human construct? for example, multiplication comes before summation, but says who? obviously it works well that way, but humans created the system.
mind you, these are very vague memories of about 4 years ago, so im sorry for the lousy wording recreation. thanks for your input. i'd like to hear from those who are good at the subjects in discussion especially 
 
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Belgium9947 Posts
The 'arguments' you bring in are kind of bad.
A function being undefined doesn't mean it's a flaw of the system.
As for multiplication coming before summation, that's just a notational agreement mathematicians made so notation would be easier. If it would be the other way around, nothing in math would change, except most formulas would need more parentheses.
If you want to argue against the scientific models as a representation of general truth in our world, you should look in to the more concrete 'postmodern' evidence of this, which is obviously a bit more complicated.
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i hate maths and physics especially measurements which has lots of types,cant they just use CM only? using inches or foot/feet makes me hard to measure.
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science never claimed to prove anything as truth.
Anything that is proven through science is always considered flawed, you can always improve it. If no major problems are found you can always refine the way it is measured or make it easier to use or easier to understand etc.
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In mathematics we start with some assumptions and prove things using them. There is no approximation to "truth". Everything you prove in mathematics is a logically true statement.
In physics we have more approximations, but I don't really know much about physics so I'll leave this alone.
x/0 being undefined means exactly that: it is undefined. That means, we have not defined what we want it to mean. x/0 is just a symbol, and we could define it to mean whatever we want and see where it gets us. From an algebraic stand point we think of this as x*(the inverse of 0). Where the inverse of 0 is some thing such that when we multiply it by 0 we get 1. Now no such thing exists in our common notion of numbers, which is why x/0 is undefined. We could invent something that has this property and throw it in and see where it takes us. The problem is that(as far as I know) it doesn't lead to anything interesting.
I think people think of mathematics in wrong way. It isn't some fixed thing. If you don't like the way something works you can just change it and get something different. That may or may not be a useful/interesting thing to do, and there is the real question.
And what does it meant to be able to prove everything? Like every truth in the universe? What is a truth? This is going to go to philosophy which I am definetly not qualified to discuss haha.
As a matter of fact however, given as set of axioms there will alwyas be a statement that can neither be proved true or false using those axioms. This was proved by Gudel, a mathematicial and logician, in the 30's I think it was. We have encountered a few of these statements, famously one that is now called the axiom of choice. People figured that it was something they wanted to be true so it is common assumed as an axiom by most (not all) mathematicians.
So, can math prove everything? Not even in its own little world..
Hopefully some of this made sense.
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Math is not a human construct, its an ultimate truth whose answer is defined by its meaning. Kind of like "I think, therefore I am".
As for dividing by zero, I always thought of it as trying to see how many empty cups of water it would it take to fill a bucket with water.
Physics is all theory and no matter how strong the evidence is you can't really prove anything.
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It seems you're too caught in the notation of math. When your say multiplication comes before summation, it's just a convention to avoid writing too many parentheses. You can pick either multiplication before addition, or addition before multiplication. Depending on which you choose, you avoid having to write parentheses in that case. More importantly, it doesn't say anything about math. The essence, or core, of math is unchanged, you're just complaining about notation.
A similar thing holds for the first bullet point. It's not surprising to me that when you divide a number by zero, you don't get a number. It happens that when you add, subtract, or multiple two numbers, you always get a number, but consider the logarithm. When you take the log of a positive number, you get a number. However, taking the log of zero or a negative number does not result in a number.
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Math and physics are very different from each other.
Physics is empirical, whatever physicians say, you always have to take into account that it could be completely wrong. Apples might fall upwards tomorrow. Physicians do know that and don't make any claims about truth. But, assuming that things stay as they are, physics provide a very precise model of reality.
Mathematicians use a given set of logic operations on a given set of axioms. Everything that is built upon that makes perfect sense and is true in the given system of logic and axioms. Unless someone manages to prove a contradiction within that system, since until now nobody was able to prove that there are no contradictions (as far as I know).
Division is well defined function. Not every function has to be defined on R.
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On December 01 2009 22:51 UGC4 wrote:
- if math/physics is such accurate systems, why do they have such innate flaws? for example, x / 0 is undefined, etc.
I don't know where to start here I'll be extremely brief. Firstly you are absolutely wrong in describing "x/0" as an innate flaw. This is not a 'flaw' of mathematics it is a mere happening, something which simply exists. Which leads me to the next point you made.
On December 01 2009 22:51 UGC4 wrote:
- why would math/physics be any more special or accurate when they are nothing but a human construct? for example, multiplication comes before summation, but says who? obviously it works well that way, but humans created the system.
What do you *mean* by special or accurate. It doesn't 'work well' that way, this was a finding, a logical pattern that people saw which simplifies multiple additions to one multiplication function.
The whole foundation of mathematics is based off very fundamental axioms, which you could argue are human constructs/assumptions, but that's about the only assumptions we ever make in the existence of being human. Everything else mathematics is, an exact city built off these axioms.
Mathematics is just a language, that's all it is. It is a langauge that one can use to describe something in utmost (quantifiable) detail.
Example,
The leaves on the trees were blowing in unison with the wind, the stem of one leaf however, was too weak to stay on the branch and thus was caught by the wind when all of a sudden a bird came swooping down with tremendous speed to eat it"
This is an English sentence we are describing things qualitatively.
All mathematics is, (in the context of physics now) is just a language to accurately describe what i s happening here so scientists and people who seek the truth to write out what this means in mathematics.
So the leaves are connected to the tree branches where the current wind speed is Wspeed = 42knots. Travelling in a direction NE 47degrees.
Leaf = Leaf y was connected on to its branch with a departing force of x Newtons. A Wspeed of Xknots produces a force of roughly X*Y Newtons, where Y is a constant which I calculate in controlled experiments. Because WspeedOnLeaf(Newtons) is > HoldStrengthOfLeaf the leaf disconnects and a bird travelling in the motion of bla
bla
bla
Do you see my point. This is mathematics. The 'flaws' in it are only a fault of the person writing the story.
When you want to argue something as intricate and detailed as mathematics you have to be extremely precise about your definitions and strong on your point of argument.
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Physics is not completely empirical, since it both begins and ends with idealized systems, rather than descriptive inferences. However physics, unlike math, cannot be a completely a priori system of knowledge, confirmed by its own definitions.
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On December 01 2009 23:08 ViruX wrote:
Math is not a human construct, its an ultimate truth whose answer is defined by its meaning. Kind of like "I think, therefore I am".
As for dividing by zero, I always thought of it as trying to see how many empty cups of water it would it take to fill a bucket with water.
Physics is all theory and no matter how strong the evidence is you can't really prove anything.
I disagree, math is a human construct. It is the tool we use to work with physics, and physics is the ultimate truth.
When i say the ultimate truth, it might sound wrong, but i really believe that if we know the starting conditions of everything (This is not actually possible as 'proven' by heisenberg) we can predict everything that follows.
Now to get back on track: Math is true. It can only be true because we defined every last bit of it as true. But you can get nothing out of math unless you apply it to a physical system.
Ehr... if this makes no sense i appologize.
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Addition comes before multiplication because multiplication is iterated addition. Multiplying two numbers x * y is calculated by adding x to itself y times. In the same way then, is raising by powers: x^y is nothing but x multiplied by itself y times, or x added to itself x times y times. Therefore, because multiplication inherently depends on addition, you cannot teach or learn multiplication before addition.
x/0 being undefined is just another truth. Tell you what, I can apply formal logic to this one. Let's assume that you can divide by zero. Then you can divide something into zero equal parts. If you have zero equal parts, then you have nothing. however, since you started with something, you obviously didn't have nothing, so by contradiction, division by zero is undefined.
EDIT: You want to talk about flaws, how about pi? The ONLY time the diameter of a circle is *ever* used in mathematics is in the definition of pi. Every other formula uses the radius. So why doesn't pi? In fact, the original definition of pi was the ratio of a circle's circumference to it's *radius* but somehow it got changed to the diameter. Those math people who might be interested in this sort of thing should read this article (PDF download).
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A more philisophical approach would be: Everything in life is defined by you. There is no ultimate truth except what you yourself perceive as true. Is an apple really an apple? What if you want it to be an mammoth? Then to you, it is a mammoth. The only problem is that in your mind you have stored the specifications of something being a apple, and something being a mammoth, so you cannot just say to yourself that something is not an apple when it clearly meets all the apple requirements..
Sorry.. i'll let myself out.
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Math starts with axioms and uses them to prove theorems, using accepted logical operations. This forces you to examine your fundamental assumptions more closely than other endeavors. There is no TRUTH with capital letters in math, only increased transparency.
As rage and others mentioned, your examples are suspect. Indeterminate does not mean "wrong" or "false" - it just means that particular operation cannot yield you a simple answer with the set of assumptions you have made. This happens often in life, even more than in math I would say! Note that changing your assumptions will make it possible to define division by zero: in abstract algebra you can extend the real numbers (or other commutative rings) into a "wheel" version, where division by zero is permitted. A bit more here.
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I was going to write a post but it proved overhelming. Instead I recommed you some books if topic interests you:
Against Method - Paul Feyerabend. About how physics is done and its relation to philosophy of science.
Philosophical papers I: Realism, Rationalism And Scientific Method - Paul Feyerabend. Clearly less fun to read than Against Method or any other on the list.
Laboratory Work: Construction Of Scientific Facts - Bruno Latour. Controversial classic (featuring amusing chapter about anthropologist who visits a strange tribe with peculiar mythology: the scientists.).
Constructing Quarks - Andrew Pickering. Less controversial than Latour, but requires more knowledge of physics to be interesting read.
IProofs and Refutations - Imre Lakatos. Concerning mathematics.
I also recommed that you borrow like a one book of Popper to go with these. When you feel head explosion coming, just spend 15 minutes with Popper.
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This "math is an invention or a discovery" debate has been going on since the dawn of time with no conclusion, not that that its usefulness would transcend the philosophical.
And I wouldn't call physics the ultimate truth. Because it changes constantly. First we used our senses to measure nature, then we used an optical microscope, then an electron microscrope. The "truth" changed in every case.
And well even though science is "flawed" with it's axioms and whatever, it may be only because we as humans are flawed: we don't have perfect senses, perfect intuition or even a perfect language, and on the latter one, it may be so flawed that some questions that can be formulated are not even worth answering. The most exemplary being "ultimate questions" such as the ones you're asking.
Though, if you compare to the alternatives to finding "ultimate truth" such as religion, the great part about science is that it can be proven wrong.
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On December 01 2009 22:51 UGC4 wrote:
- why would math/physics be any more special or accurate when they are nothing but a human construct? for example, multiplication comes before summation, but says who? obviously it works well that way, but humans created the system.
I reread this after I wrote my previous post. Did you mean in the order of operations? Obviously I assumed you meant when basic math is taught, but in that case addition is usually taught before multiplication, so the only thing I can think of is the order of operations, in which case I have an explanation that is more to your question.
In order to fully explain the question, let's start with the beginning of the order of operations. I'll spoiler it for those who are interested.
+ Show Spoiler +For reference, the order of operations are:
- Parentheses (functions like log, ln, sin, etc usually imply parentheses as well)
- Exponents
- Multiplication or division
- Addition or subtraction
The reason parentheses are first is because they are explicit in telling someone to (calculate what's inside me first!) It's essentially the mathematician's override button for the order of operations.
Multiplication and division are exactly the same operation, if you think about it in the sense that division is multiplication of the reciprocal, and in the same way addition and subtraction are the same if you think about subtraction as addition of the opposite. This is why multiplication and division are on the same step, and addition and subtraction are on the next step.
Now, the reason the order is Exponents, Multiplication/Division, Addition/Subtraction is based on the reasons I outlined in my previous post. Multiplication is nothing more than iterated addition, so when we have a more complicated operation, you can think of multiplication as shorthand for more addition, the same way exponentials are shorthand for more multiplication.
So, I think what you're asking here is, if we had the equation w + x * y, why do we assume w + (x * y) instead of (w + x) * y? The reason is multiplication is really just addition anyways, so when you evaluate the equation, you're really just rewriting it w + x + x + x... until you had enough x's.
In short, All of the basic mathematical operators only use the numbers directly before and directly after, unless we use parentheses explicitly stating otherwise.
Also, I'd like to argue your point that humans created the system. We didn't really create it, the universe did. As much as I hate using kindergarten examples, if I have two apples and I give you one, it's not a human construction that I only have one apple left. You could argue that the notation is a human construct, and it is, but the abstractions it represents is not.
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I'm insulted that you would include physics in the same breath as maths when considering their logical purity.
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X / 0 is definately not undefined. It's absolutely impossible to spread a finite value of X over a negative infinite space, in the 3 dimensions as we know them.
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There is very little truth claimed by math and physics. That makes it pure. It's like in poker, if you know someone plays very little hands, and whenever he plays a hand he has the cards to back it up, he is a solid player. Same goes for math + physics. Since the amount of truth they claim is so few, whenever it claims something, it's very likely to be true.
To me, math is creating world with a set of rules. (like sc is a world with a set of rules). Mathematicians create, and have created, a lot of different worlds. Phyicisists take various of those worlds and apply the accompanied set of rules to their real world problem. If the mathematic world behaves the same as the real world, they keep those set of rules. That way they can predict behaviour in the real world by calculating it in their mathematical world.
You could create a world where x/0 is defined. I did that with a friend. I was like "why is -1^(1/2) = i? I claim that x/0 = p." But what happened is that in the world where x/0 = p, it is also true that -1 = 1. This is not a very useful world. You can't apply it to any real world situations I can think of.
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On December 01 2009 22:51 UGC4 wrote:Hi everyone, before you read on let me warn you, this will probably be a boring blog to some of you because it is math related. i dislike math myself, even though i know how useful it can be. i just wanna discuss a couple of aspects of it with the smart TL community that cares enough to read. Anyway. I remember having a discussion in a class regarding the topic of TRUTH. The discussion then derailed into the subjects of Mathematics + Physics and their proximity to perfection or truth, because most students argued that these sciences could prove everything. I understood their argument completely for obvious reasons, but then i asked the following: - if math/physics is such accurate systems, why do they have such innate flaws? for example, x / 0 is undefined, etc. - why would math/physics be any more special or accurate when they are nothing but a human construct? for example, multiplication comes before summation, but says who? obviously it works well that way, but humans created the system. mind you, these are very vague memories of about 4 years ago, so im sorry for the lousy wording recreation. thanks for your input. i'd like to hear from those who are good at the subjects in discussion especially
Mathematical truth is strange. Here's something I am currently annoyed at:
Suppose you have a sound system of logic L, that is, all theorems of L are true. Now suppose L is also powerful enough to prove all true statements of the form "Turing machine M halts on input x" which can be done by simulating M until it halts. This is NOT the halting problem.
Let Tj be the Turing machine which on input i halts if the statement "Ti fails to halt on input i" is provable in L, and otherwise does not halt. Then the statement "Tj fails to halt on input j" is true but not provable in L.
This is because if Tj halted on j, by definition of Tj it follows that Tj does not halt on j. Since L is sound, the latter must be true, thus a contradiction. Therefore Tj does not halt on input j. Again by definition of Tj, it follows that it is not provable in L that Tj does not halt on input j.
But yet we have determined that Tj does not halt. Therefore we do not think within any fixed sound logical system, no matter how vast it is.
What then is our logic?
oops there were a few typos ^^
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I think you're talking about the Halting problem.
It's basically a proof that there exist some problems that cannot be solved by computers, no matter how cleverly designed.
More examples of uncomputable problems:
A barber shaves everyone who doesn't shave himself. Does the barber shave himself?
The adjective "heterological" describes adjectives which do not possess the trait they describe. Is the word heterological heterological?
A set S is defined as containing every element x that is not in S. What elements are in set S?
Now, get your head around this:
There are an uncountably infinite number of problems (larger than the set of computable problems for which there exists a program to solve it).
For every computable problem, there exists an infinite number of programs that can solve it (just add blank spaces or comments as many times as you like, and you have an infinite number of variations of the same program).
However, the number of programs that exist are countably infinite, which is infinitely smaller than the number of problems that exist. (Proof that programs are countably infinite: map them by lexographical order to the natural numbers)
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Nah it's not the halting problem. It's Gödel's incompleteness theorem in the context of Turing machines. Or rather, it is how you would generate such an undecideable problem.
There are an uncountably infinite number of problems (larger than the set of computable problems for which there exists a program to solve it).
For every computable problem, there exists an infinite number of programs that can solve it (just add blank spaces or comments as many times as you like, and you have an infinite number of variations of the same program).
However, the number of programs that exist are countably infinite, which is infinitely smaller than the number of problems that exist. (Proof that programs are countably infinite: map them by lexographical order to the natural numbers)
Not sure how the first paragraph is relevant to the next two (since there are a countably infinite number of computable problems). Seems to just be an analogy of how every integer appears in an infinite number of rational numbers, but both are countably infinite sets.
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On December 02 2009 01:05 EtherealDeath wrote:Show nested quote +On December 01 2009 22:51 UGC4 wrote:Hi everyone, before you read on let me warn you, this will probably be a boring blog to some of you because it is math related. i dislike math myself, even though i know how useful it can be. i just wanna discuss a couple of aspects of it with the smart TL community that cares enough to read. Anyway. I remember having a discussion in a class regarding the topic of TRUTH. The discussion then derailed into the subjects of Mathematics + Physics and their proximity to perfection or truth, because most students argued that these sciences could prove everything. I understood their argument completely for obvious reasons, but then i asked the following: - if math/physics is such accurate systems, why do they have such innate flaws? for example, x / 0 is undefined, etc. - why would math/physics be any more special or accurate when they are nothing but a human construct? for example, multiplication comes before summation, but says who? obviously it works well that way, but humans created the system. mind you, these are very vague memories of about 4 years ago, so im sorry for the lousy wording recreation. thanks for your input. i'd like to hear from those who are good at the subjects in discussion especially Mathematical truth is strange. Here's something I am currently annoyed at: Suppose you have a sound system of logic L, that is, all theorems of L are true. Now suppose L is also powerful enough to prove all true statements of the form "Turing machine M halts on input x" which can be done by simulating M until it halts. This is NOT the halting problem.
Does this include all true statements of the form "Turing machine M does not halt on input x?" Also, does this also mean it is strong enough to disprove all false statements of the same forms?
Let Tj be the Turing machine which on input i halts if the statement "Ti fails to halt on input i" is provable in L, and otherwise does not halt. Then the statement "Tj fails to halt on input j" is true but not provable in L.
You don't mention Ti again in your post. Is Tj supposed to be a recursive Turing machine, or does it work on a different Turing machine?
This is because if Tj halted on j, by definition of Tj it follows that Tj does not halt on j. Since L is sound, the latter must be true, thus a contradiction. Therefore Tj does not halt on input j. Again by definition of Tj, it follows that it is not provable in L that Tj does not halt on input j.
But yet we have determined that Tj does not halt. Therefore we do not think within any fixed sound logical system, no matter how vast it is.
What then is our logic?
oops there were a few typos ^^
I need to know the answers to my previous questions before I can continue my response. Thanks for the intriguing puzzle though!
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On December 02 2009 02:21 EtherealDeath wrote:Nah it's not the halting problem. It's Gödel's incompleteness theorem in the context of Turing machines. Or rather, it is how you would generate such an undecideable problem. Show nested quote +
There are an uncountably infinite number of problems (larger than the set of computable problems for which there exists a program to solve it).
For every computable problem, there exists an infinite number of programs that can solve it (just add blank spaces or comments as many times as you like, and you have an infinite number of variations of the same program).
However, the number of programs that exist are countably infinite, which is infinitely smaller than the number of problems that exist. (Proof that programs are countably infinite: map them by lexographical order to the natural numbers)
Not sure how the first paragraph is relevant to the next two (since there are a countably infinite number of computable problems). Seems to just be an analogy of how every integer appears in an infinite number of rational numbers, but both are countably infinite sets.
If I understand them correctly, all those examples are Godel's Incompleteness Theorem, just different examples of it.
@OP Alot of those things are due to the axioms that mathematicians choose to work with.
There have been some experimentation in other axiom sets that produce some really weird results, but those results are largely dismissed by the mathematic community.
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On December 01 2009 23:08 ViruX wrote:
Math is not a human construct, its an ultimate truth whose answer is defined by its meaning. Kind of like "I think, therefore I am".
As for dividing by zero, I always thought of it as trying to see how many empty cups of water it would it take to fill a bucket with water.
Physics is all theory and no matter how strong the evidence is you can't really prove anything.
That's a really cool way of thinking about the divide by zero problem...
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Does this include all true statements of the form "Turing machine M does not halt on input x?" Also, does this also mean it is strong enough to disprove all false statements of the same forms?
Sure, that can be in L. And no, L is not necessarily capable of disproving all false statements, or any at all, so we assume nothing about its power in this regard.
You don't mention Ti again in your post. Is Tj supposed to be a recursive Turing machine, or does it work on a different Turing machine?
Yep, you put Tj into itself (so that Tj is the Ti mentioned in the general functioning of Tj).
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There's a pretty amusing paper on this from the 1960s, regarding consistency and various things:
Minds, Machines, and Godel J.R.Lucas
+ Show Spoiler [An excerpt] + If such a machine were built to produce theorems about arithmetic (in many ways the simplest part of mathematics), it would have only a finite number of components, and so there would be only a finite number of types of operation it could do, and only a finite number of initial (115) assumptions it could operate on. Indeed, we can go further, and say that there would only be a definite number of types of operation, and of initial assumptions, that could be built into it. Machines are definite: anything which was indefinite or infinite we [258] should not count as a machine. Note that we say number of types of operation, not number of operations. Given sufficient time, and provided that it did not wear out, a machine could go on repeating an operation indefinitely: it is merely that there can be only a definite number of different sorts of operation it can perform.
If there are only a definite number of types of operation and initial assumptions built into the system, we can represent them all by suitable symbols written down on paper. We can parallel the operation by rules ("rules of inference" or "axiom schemata") allowing us to go from one or more formulae (or even from no formula at all) to another formula, and we can parallel the initial assumptions (if any) by a set of initial formulae ("primitive propositions", "postulates" or "axioms"). Once we have represented these on paper, we can represent every single operation: all we need do is to give formulae representing the situation before and after the operation, and note which rule is being invoked. We can thus represent on paper any possible sequence of operations the machine might perform. However long, the machine went on operating, we could, give enough time, paper and patience, write down an analogue of the machine's operations. This analogue would in fact be a formal proof: every operation of the machine is represented by the application of one of the rules: and the conditions which determine for the machine whether an operation can be performed in a certain situation, become, in our representation, conditions which settle whether a rule can be applied to a certain formula, i.e., formal conditions of applicability. Thus, construing our rules as rules of inference, we shall have a proof-sequence of {47} formulae, each one being written down in virtue of some formal rule of inference having been applied to some previous formula or formulae (except, of course, for the initial formulae, which are given because they represent initial assumptions built into the system). The conclusions it is possible for the machine to produce as being true will therefore correspond to the theorems that can be proved in the corresponding formal system. We now construct a Goedelian formula in this formal system. This formula cannot be proved-in-the- system. Therefore the machine cannot produce the corresponding formula as being true. But we can see that the Goedelian formula is true: any rational being could follow Goedel's argument, and convince himself that the Goedelian formula, although unprovable-in-the-system, was nonetheless----in fact, for that very reason---true. Now any mechanical model of the mind must include a mechanism which can enunciate truths of arithmetic, because this is something which minds can do: in fact, it is easy to produce mechanical models which will in many respects produce truths of arithmetic far [259] better than human beings can. But in this one respect they cannot do so well: in that for every machine there is a truth which it cannot produce as being true, but which a mind can. This shows that a machine cannot be a complete and adequate model of the mind. It cannot do everything that a mind can do, since however much it can do, there is always something which it cannot do, and a mind can. This is not to say that we cannot build a machine to simulate any desired piece of mind-like behaviour: it is only that we cannot build a machine to simulate every piece of mind-like behaviour. We can (or shall be able to one day) build machines capable of reproducing bits of mind-like behaviour, and indeed of outdoing the performances of human minds: but however good the machine is, and however much better (116) it can do in nearly all respects than a human mind can, it always has this one weakness, this one thing which it cannot do, whereas a mind can. The Goedelian formula is the Achilles' heel of the cybernetical machine. And therefore we cannot hope ever to produce a machine that will be able to do all that a mind can do: we can never not even in principle, have a mechanical model of the mind.
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Okay, so let me go over the problem again. Tj is a recursive Turing machine that halts if and only if the statement "Tj fails to halt on input j" is provable in L. Then it follows that if Tj does not halt, then the statement "Tj fails to halt on input j" is not provable in L.
If Tj halted on j, then the statement "Tj fails to halt on input j" is false. In our definition, we said L is strong enough to prove all true statements of this form, but not strong enough to disprove a statement. So unless L is strong enough to prove a false statement is true, then I don't see any contradiction in logic here.
I think it might be how you worded the problem, but the definition you gave for Tj is: Let Tj be the Turing machine which on input i halts if the statement "Ti fails to halt on input i" is provable in L, and otherwise does not halt. So the issue isn't whether Tj halts or not, but whether it is provable in L.
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All of these uncomputable problems I listed (which I'm sure are related to the discussion on logic systems) have in common the characteristic of denying their own answers. I'm not sure if there is any other kind of uncomputable problem.
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On December 02 2009 03:21 vAltyR wrote:Okay, so let me go over the problem again. Tj is a recursive Turing machine that halts if and only if the statement "Tj fails to halt on input j" is provable in L. Then it follows that if Tj does not halt, then the statement "Tj fails to halt on input j" is not provable in L. If Tj halted on j, then the statement "Tj fails to halt on input j" is false. In our definition, we said L is strong enough to prove all true statements of this form, but not strong enough to disprove a statement. So unless L is strong enough to prove a false statement is true, then I don't see any contradiction in logic here. I think it might be how you worded the problem, but the definition you gave for Tj is: Show nested quote +Let Tj be the Turing machine which on input i halts if the statement "Ti fails to halt on input i" is provable in L, and otherwise does not halt. So the issue isn't whether Tj halts or not, but whether it is provable in L.
So we have Tj(j) halts if "Tj fails to halt on input j" is provable in L, and otherwise does not halt.. If Tj halts, then the statement in parentheses "Tj fails to halt on input j" must be provable in L by defintion of Tj. Since L is sound, it follows that this must be true, that Tj fails to halt on j, contradicting the assumption that Tj halts on j. Therefore Tj does not halt on j. By definition of Tj(j) not halting, this means that in L it is not provable that Tj does not halt on input j. However, we have just shown that it is in fact true that Tj does not halt on j. Thus we have a statement that we know to be true but nevertheless is not provable within L, regardless of what L is, so long as L is a sound system of logic and is powerful enough to prove all true statements of the form "Turing machine M halts on input x", but the latter condition is pretty obvious for humans: just simulate M. Thus it seems we are left wondering whether humans reason with any fixed sound system of logic.
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wow, thanks everyone you guys consistently exceed my expectations when it comes to input.
i'd like to quickly apologize again for the bad wording of the discussion, i know "truth" and "perfect" are impossible to define but im glad most people got my point.
i should also apologize for the lousy examples that i provided for my questions, seen as they do seem to have an answer. but it was good to see other people put forward their own examples, which i guess leaves the ultimate question of whether math is a human construct or not unanswered, but im perfectly fine with that. thanks again for all your input, and for the recommended reads. i'll be sure to take a look at them!
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On December 02 2009 04:50 UGC4 wrote:wow, thanks everyone you guys consistently exceed my expectations when it comes to input. i'd like to quickly apologize again for the bad wording of the discussion, i know "truth" and "perfect" are impossible to define but im glad most people got my point. i should also apologize for the lousy examples that i provided for my questions, seen as they do seem to have an answer. but it was good to see other people put forward their own examples, which i guess leaves the ultimate question of whether math is a human construct or not unanswered, but im perfectly fine with that. thanks again for all your input, and for the recommended reads. i'll be sure to take a look at them!
Oh and as for your question of whether math is a human construct or something, I suppose the real question is do the objects of mathematics reside anywhere. As its constructs are infinite, it is impossible to store them anywhere in the physical universe (not enough particles!). How would you create a perfect line? Not enough atoms to do it, and besides atoms aren't small enough; nevertheless, the line obviously exists right? And then it goes into random philosophical bs, but yea -_- (the world of the mathematical objects anyone?)
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Math has no flaws. Even things people think are "flaws" like the Banach-Tarski "paradox" aren't really flaws (nor is it really a paradox -- it's just called that because it defies intuition).
However, math is something we create. We take a bunch of statements that are true, create new definitions and then use things from the set of discovered truths to create new truths. If anything, what's so stunning about all of this is that these new truths that get uncovered turn out to have vast application to the "real world," even if no such intention was ever at work.
Physics, however, may be very "truthy," but is not "truth." It's an attempt to explain observable phenomena in the universe, but we have no way of knowing if any of what we've come up with is actually "the truth." We merely suspect that it is because our physical theories have withstood the test of time. But that doesn't mean they are "right."
Physics just tends to come out "better" than other forms of science because it's far less politicized. When a biologist does research on the genome, he's under political pressures he may not even comprehend from various groups with various objectives. Even in the United States, there was an intellectual movement advocating euthanasia of people with "unfavorable genes," that only lost steam after the world witnessed the horrors of the Holocaust. And even today, evolution can be a dangerously touchy subject.
And don't even get me started on the so-called "social sciences," where its easy to get ostracized if you present evidence that goes against the status quo.
In the end, just remember:
Biologists think they are chemists, chemists think they are physicists, physicists think they are god, and god is a mathematician.
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EPT
