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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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Belgium9942 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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EPT
