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> what is math?, no joke
L3ma
Posted: Aug 3 2005, 04:03 PM
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So what is math? Is it how we see the world? Will 2 + 2 always equal 4?
It seems to be the nature of math that when an equation is executed it will be the same every time. Our math is formed from serial thinking. If our minds were capable of parallel thinking then math problems might not have the same outcome every time. Math might become more like spoken language with the meanings always changing. In our minds we see the basic integers as absolute, but so few things in the universe are. Perhaps the concept of absolute integers is too limiting.
The idea of integers as we know them today came from the Babylonians (I think). Before that, math of the time was not capable of multiplication. Modern math seems to be the best way to explain the universe but suppose someday we discover something that does not fit with math as we know it? We would have to abandon modern math and invent something new. If math is a concept then it isn't absolute. 2+2 equals 4 simply because you say so. There is nothing stopping you from saying 2 plus 2 equal’s fish. Of course putting 2 stones with 2 stones won't create 5, but suppose scientists witness something that breaks the rules we know? Then the rules must be wrong and we might discover that the statement 2+2=4 is only true under certain circumstances.
Scientists think that the universe may continue to expand into nothingness. This happens then as the universe decays the laws as we know them will slowly warp. Math may then change. 2+2 may then equal infinity. You cannot disprove this with math that is subject to modern laws of the universe in a time when those laws are no longer in effect. This is just a hypothetical example for showing what I mean. In reality the universe will definitely change because our knowledge of it will change. Will mathematics change then? Is there anything of mathematics that is absolute?
Sorry if my info is scattered. I’m no Shakespeare
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UserGoogol
Posted: Aug 3 2005, 04:23 PM
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Mathematics is basically a logical construct. It's not so much that 1 + 1 = 2 because we say so, but other things in math are "because we say so." Math is not science, it doesn't have to describe anything, it just has to be logically consistant. (Although math was invented to describe the universe and it so happens that it has been very good at that task.)

I had a proof that 1 + 1 = 2, from the Peano Axioms and the definitions of 1, 2, and addition, but I thought I was glossing over some things so I decided to edit it out.

This post has been edited by UserGoogol on Aug 3 2005, 04:28 PM
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Umino
Posted: Aug 3 2005, 04:26 PM
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Your theories are not that far from the truth, l3ma. I recently discovered that the multiplication of two negatives results in another negative, despite what your math-books says. Take -3 and -5 in example.

-3 × -5 = (-1 × 3) × (-1 × 5) = -1 × (3 × 5) = -1 × 15 = -15

omg.
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Shaw
Posted: Aug 3 2005, 04:58 PM
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QUOTE
-3 × -5 = (-1 × 3) × (-1 × 5) = -1 × (3 × 5) = -1 × 15 = -15


-3*-5=(-1*3)(-1*5)DNE-1*(3*5)

-3*-5=(-1*3)(-1*5)=(-1*-1)(3*5)=15


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Loismustdie
Posted: Aug 3 2005, 07:00 PM
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OP: I would argue that you are very very wrong.
To me, the only absolute in the world is mathmatics. Philisophicaly speaking, one can't prove ones own existance to another person, yet no matter how one looks at it X=X is fact. If by 2X we mean X+X, then 2X > X, again another fact.

Yes we can play with numbers and call them whatever we like, we can even change the base we count with using hexidecimal or binary, but all that does is change the method we use to see the numbers, they still mean the same thing.

Basic maths (addition, subtraction, multiplication, division) is pure logic, logic that in all instances is true. Now I'm not denying that maths is flawed, as things get more complecated we can see these flaws, but in its basic form maths is logic of the purest form.

EDIT: Umino - Wrong. Negative X Negative is the same as Positive X Positive, in that it always ends with a positive. Negative X Positive and Positive X Negative are what you need to get negative result.

This post has been edited by Loismustdie on Aug 3 2005, 07:04 PM
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I'm Unemployed
Posted: Aug 3 2005, 07:37 PM
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QUOTE (L3ma @ Aug 3 2005, 03:03 PM)
Will 2 + 2 always equal 4?

Yes, the concept is objective and permanent. Although the language referring to it may change, the concept itself supercedes humanity. Partly because the concept is rooted outside of physical existence; it's universal and abstract.

Mathematical concepts have always existed, since the first millisecond of the universe's existence. What changes is people's ability to comprehend and calculate. 2+2=4 exists regardless of what exists physically in the universe, because you don't need anything in order to calculate 2+2=4. Just like how a sign can be true if it says "All trespassers will be shot" even if there are no trespassers, it's true that 2+2=4 even if there is nothing in existence. What is represented by 2 or 4 need not be instantiated for 2+2 to equal 4.

This post has been edited by I'm Unemployed on Aug 3 2005, 07:40 PM
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L3ma
Posted: Aug 4 2005, 02:46 AM
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it's true that 2+2=4 even if there is nothing in existence

so why does 2+2=4? and how? if nothing existed than you wouldn't have two pebbles to put with two more to call four. the idea of 2 and four wouldn't even come to you. in contrast we live in a universe where everything exists, or more than we know of. we will never know everything. as we learn more our ideas will change. think about it. in a universe with nothing there is no math. in a universe with everything, nothing is absolute. or is it? if two pebbles wash up on the beach beside two others, you get four right? actually no, no unless there there is a person there to observe it and call it four. that's how i see it anyway.
math will evolve and change. the concept of zero hasn't always been around, neither has infinity. we can expect the inconceivable. we might invent something higher than infinity. someday the concept of 2 and 4 might become absolete like roman numerals.
maybe i'm totally off about this. if a tree falls in the forest, closing your eyes and covering your ears won't stop the tree from falling and breaking your philisophical neck.
Oh my head. XoX
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Buugipopuu
Posted: Aug 4 2005, 04:42 AM
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QUOTE (L3ma @ Aug 4 2005, 09:46 AM)
so why does 2+2=4?

Because Bertrand Russel and Alfred North Whitehead proved it in Principia Mathematica (Well, they proved 1+1=2, and it's a relatively simple matter of induction to get 2+2=4 from 1+1=2).

QUOTE

no unless there there is a person there to observe it and call it four. that's how i see it anyway


That's because you're wrong. Four still exists even if there is nobody to call it 'four'. 'Four' doesn't become something else when it's called 'Vier', 'Quatre', 'Yon', or 'Cuatro', so why should it be anything else when it isn't called anything at all?

QUOTE
the concept of zero hasn't always been around, neither has infinity


The concept has always been around. The concept can't not exist. (unless you get into the realm of metafurnaces, but that doesn't count) It hasn't always been known to humanity on the other hand.

QUOTE

we might invent something higher than infinity
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Shish
Posted: Aug 4 2005, 06:48 AM
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> no, no unless there there is a person there to observe it

What you're thinking of is philosophy, not math; it's the same question of trees falling being heard by no-one -- like much of philosophy, it isn't even scientifically valid in the first place...

--

Threads like this angst me up so much I need to blog to get rid of it, and are the reason I only come to HQR in order to make meta-posts :| (And I do hope I haven't overestimated Umino; I shall be terribly depressed if he was being serious, but for now I will assume he was being subtly witty which nobody understood...)
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The_Cat
Posted: Aug 4 2005, 10:51 AM
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If you take the infinity of the universe as an example, you're talking limits. You could state that 2 + 2 doesn't equal 4, but 4 +/- an infinitely small number.

But in the end, that's just screwing around and making a lot of effort in stating the obvious.
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stuff
Posted: Aug 4 2005, 12:21 PM
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QUOTE (The_Cat)
You could state that 2 + 2 doesn't equal 4.

Only if you really enjoy making false statements.
QUOTE (The_Cat)
but 4 +/- an infinitely small number.

The use of the word "but" should be reserved for actual contradictions only.

(4) equals (4 +/- an infinitely small number).


Thanks, now I feel bad about letting this thread live a few seconds longer. sad.gif
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I'm Unemployed
Posted: Aug 4 2005, 01:59 PM
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QUOTE (Shish @ Aug 4 2005, 05:48 AM)
like much of philosophy, it isn't even scientifically valid in the first place...

That's a pretty funny statement, considering that philosophy was once a branch of science, philosophers oftentimes double as empirical scientists, and, today, philosophy more than often coincides with science (they aren't mutually exclusive). Science is merely correct observation. Philosophy is merely correct reasoning/thinking. There's nothing keeping the two exclusive of each other.

And it's not really philosophy that he's talking. It's a logical truth that some concepts, such as math, need not be instantiated for them to be true. In such a case, the personal belief that observations need to be observed in order to be true is false, logically and practically.

This post has been edited by I'm Unemployed on Aug 4 2005, 02:01 PM
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dark_deva
Posted: Aug 4 2005, 02:27 PM
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The statement 2+2=4 seems to be a favorite target of those bent on showing math does not equal truth. There is a linguistic problem intimately bound up with all such discussions, which is subtle enough that most people miss it entirely. Simply put, the statement "2+2=4" isn't actually a true statement, but refers to a true statement that cannot adequately be expressed in words. It is the referenced statement that is so tediously proven in Principia, and even Russell realized that it was not possible to adequately define the terms he was using. What is commonly meant by 2+2=4 is intrinsically true, independent of the imprecise methods of description of this fact. The flip side of Godel's Incompleteness Theorem (which essentially says that any logical system of sufficient complexity is unable to decide certain questions) is that by suitably restricting the scope of application of a logical system you can force it to be consistent. The Peano axiomatic system (often referred to somewhat imprecisely by ordinary people as "numbers", usually with emphasis on the "numb") is one such system. Euclidean geometry is another such system.
A separate yet related issue lies in the confusion of the (abstract) mathematics with the realm of application of the mathematical system. People often think of the number "2" for example as 2 things: 2 cars, 2 billliard balls, 2 oranges, 2 pennies, whatever. The concept of 2 has nothing to do with cars, or billiard balls, orange or pennies. It is an abstraction of a property of all sets with two things in them, just as unity (1) and nothing (0) are. Because it is an abstraction, it does not behave in quite the same way as tangible physical objects, although sometimes you are able to talk about it as if it actually has physical existence (I can show you "twoness" with 2 pennies).
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Loismustdie
Posted: Aug 4 2005, 03:21 PM
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QUOTE (dark_deva @ Aug 4 2005, 02:27 PM)
Simply put, the statement "2+2=4" isn't actually a true statement, but refers to a true statement that cannot adequately be expressed in words.

A statment is the end product of someone stating something, right?
The fact is 2+2=4, me writing this is the statment. "2+2=4" does not /refer/ to a statment, it is a statement refering to the /fact/ that 4 is made of two lots of 2, or however you want to put it.

I fail to see why that statment is not true.

This post has been edited by Loismustdie on Aug 4 2005, 03:21 PM
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rebecacaca
Posted: Aug 4 2005, 03:37 PM
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QUOTE (L3ma @ Aug 3 2005, 04:03 PM)
So what is math?

"The science of patterns"
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AllanO
Posted: Aug 4 2005, 04:32 PM
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Basically like everyone else has said mathematical propositions follow by definition, if you take all the premises of the argument and follow the rules you must come to that conclusion. The force of logical consistency is unavoidable, if any logical inconsistancy is allowed than anything might be allowed (if 1=0 then every number is equal to every other number for example) and so to try and talk about a mathematical proposition no longer following is to say imagine what follows from a contradiction (it is in a real sense unimaginable).

Also, human language is equally as meaningful and unchanging, what you meant by your question means the same thing even if 500 years ago no one could have understood it and even if 500 years from now no one will either. If 500 years from now someone digs up this post they may not understand it but if they study the English language of 2005 long enough they may well be able to do so.

In a sense mathematical statements are only valid within their proper context also. "2+2=10" is proper math if I point out that it is in a base four number system (quatrinary or quatrial?).

Physical systems may or may not have behaviour consistent with a given mathematical system but physical systems are consistent (or at least I can not imagine it is otherwise) so the conclusion that a mathematical analog to a given physical system exists seems to follow. If you mix 2 Litres of alcohol and 2 L of water you get less than 4 L of their mixture (because they disolve into each other), however every time you do such a mixture in the same temperature and pressure you get the same result. I am pretty sure some mathematical scheme exists that could tell you what the final volume of your miture would be based on the volume of alcohol and water added together.

Mathematics is a science in the sense of it being a systematic branch of knowledge, but it is not an empirical science.

On the truth of mathematics, some people might want to deny that mathematics is true while admitting it could not be otherwise. They might for example take the view that mathematical statements are true only in a similiar sense to the way that "Ping is an EDS" is a true statement (this is called mathematical fictionalism). "Ping is an EDS" is true in the sense that the webcomic Megatokyo Ping is refered to as an EDS. So someone taking this way might go "2+2=4" is true in the sense that such a statement follows from Peano's axioms plus a proper definition of 2,4,+ and =. Certainly a lot of people would want to avoid mathematical Platonism, which takes it that mathematical objects are real in some strong sense. I think though this is an issue of the nature of truth though rather than a disagreement about most of the proprities of mathematical statements. It might also reflect a difference in how one choses to describe the relation between mathematical and physical statements.

As to the topic question, I think mathematics is hard to define definitively. It is the study of systems of concepts specifically ones that follow explicitly defined rules and have explicitly defined elements. However, not all such systems are the study of mathematics (at least as it is used in normal language) because logic is such a system but studying logic is not usually viewed as studying mathematics. Also, idealized games of chess and idealized computers are such systems but are not usually thought of as math either. Since there are an infinite number of possible systems for mathematicians to study mathematics is a very open ended subject. There are some systems that mathematics is clearly concerned with the number system, geometrical systems, algebra and so on.
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Shish
Posted: Aug 4 2005, 06:13 PM
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>> like much of philosophy, it isn't even scientifically valid in the first place...
> That's a pretty funny statement, considering that philosophy was once a branch of science

So it is, my choice of phrasing was quite poor there... My point was to draw the similarities between "is 2+2 still 4 if there are no objects to count?" and "if a tree falls and nobody hears, did it make a sound?" -- what I meant by invalid is that in the case of the tree, nothing happens in isolation and can be measured scientifically, so you can't reasonably ask for both; and in the case of the numbers, 2+2 can't equal anything but 4, so you can't reasonably ask "what if 2+2 isn't 4?"

Perhaps it also would've been clearer if I'd specified stoner philosophy, where such questions are perfectly acceptable...
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Umino
Posted: Aug 5 2005, 07:15 AM
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QUOTE
EDIT: Umino - Wrong. Negative X Negative is the same as Positive X Positive, in that it always ends with a positive. Negative X Positive and Positive X Negative are what you need to get negative result.

This translates to: "Your statement is incorrect because it is not correct."

It'd have been cool if you'd proven why negative times negative equals positive. You can easily prove that negative times positive equals negative, and from that prove that negative times negative equals positive. In fact, you only have to use the axiom that A(BC) = (AB)C, that would've been a far more appropriate answer than: "wrong, because it's not right."
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Loismustdie
Posted: Aug 5 2005, 07:27 AM
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QUOTE (Umino @ Aug 5 2005, 07:15 AM)
This translates to: "Your statement is incorrect because it is not correct."

Well no, it's, 'wrong, expansion on the mattter'.
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CodeWarrior
Posted: Aug 6 2005, 07:09 PM
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[sigh] maths is the study via axiomatic logic of precisely defined, usually quantitative, statements.

2+2=4 can be proven from set theory and it's tricky. the proof I will present will not be rigourous. Clearly it's posable to have a collection of things (a set) and there must be a set containing no things the empty set. Now this set is in it's self a thing. it is one thing. So we now have the concepts of 0. we can encapsulate this empty set in another set and call it 1. we may proceed like a reverse version of pass the parcel to define each number by it's predecessor. This is essentially counting up. we can also revers this process and unwrap the parcels this is counting down. Now this is not quite enough but assume we can now count the elements in a set. We want to combine the elements of 2 sets both equivalent to the number 2 (containing 2 elements). so we take one element from the first set and put it in the second. hence we have a set equivalent to one and a set equivalent to 3. we repeat and we have one set that is empty and one that is equivalent to 4. You could call this set 10 but 10 would then be 1 more that 3 and 1 less than 5.

In fact once you have proved basic arithmetic from set theory rationals and the completeness axiom follow on rather obviously. It is entirely feasible that a paralysed deaf blind man with out taste smell or touch could be the greatest mathematician in the world having noticed he could group his thoughts in to set like structures and deriving all maths from this.

Oh and and philosophy did not grow from science. What we now call science was once called natural philosophy and was treated as a branch of philosophy. Also some would argue maths is the basis for philosophy. Indeed Plato had the words "Let no one unversed in mathematics enter here" over the main door to his academy of philosophy and was highly influenced by Pythagorus.

This post has been edited by CodeWarrior on Aug 6 2005, 07:17 PM
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seyosama
Posted: Aug 6 2005, 09:57 PM
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It helps that the greek word for mathematics came from another greek word meaning "fond of learning."

I cannot remember the actual inscription over the Lyceum, but it either directly means 'fond of learning' and mathematics by proxy, or it means mathematics directly (and vice versa).
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AllanO
Posted: Aug 7 2005, 10:09 AM
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First the saying said (I suspect it is aprocryphal) to be above the Lyceaum is "Let no one who has not studied geometry enter here." or something like that it is definetly geometry (keep in mind greek geometry is synthetic consisting of a proof proceeding from agreed upon principles through an argument to the conclusion of a theorem) and not mathematics.

Our word math comes from the greek participle for learning, this meaning survives in the word polymath (someone who knows many different things). I do not know when the narrowing of the meaning of math occurred.

I think I remember reading once that the Pythagoreans were the first to call themselves philosophers (literally lover of wisdom), but I would not be sure about that. If that is the case it gives a mathematical tradition a tenous link as the origins of Western philosophy and science. However, the term philosophy is applied to thing like Chinese scholarly traditions (Confuscianism, Taoism, etc.) that have completely different origin. So philosophy is defined conceptually rather than historically. Even under a narrow interpretation of philosophy the foundations of mathematics and science are a philosophical concern.

I found a source on the inscription here. The usual translation is "Let no one ignorant of geometry enter." The root word is geometron. It is probably apocryphal though it does suggest Plato's view that geometry is a good model of knowledge and truth.
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CodeWarrior
Posted: Aug 7 2005, 11:23 AM
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QUOTE (AllanO @ Aug 7 2005, 04:09 PM)
First the saying said (I suspect it is aprocryphal) to be above the Lyceaum is "Let no one who has not studied geometry enter here." or something like that it is definetly geometry (keep in mind greek geometry is synthetic consisting of a proof proceeding from agreed upon principles through an argument to the conclusion of a theorem) and not mathematics.

Our word math comes from the greek participle for learning, this meaning survives in the word polymath (someone who knows many different things). I do not know when the narrowing of the meaning of math occurred.

I think I remember reading once that the Pythagoreans were the first to call themselves philosophers (literally lover of wisdom), but I would not be sure about that. If that is the case it gives a mathematical tradition a tenous link as the origins of Western philosophy and science. However, the term philosophy is applied to thing like Chinese scholarly traditions (Confuscianism, Taoism, etc.) that have completely different origin. So philosophy is defined conceptually rather than historically. Even under a narrow interpretation of philosophy the foundations of mathematics and science are a philosophical concern.

I found a source on the inscription here. The usual translation is "Let no one ignorant of geometry enter." The root word is geometron. It is probably apocryphal though it does suggest Plato's view that geometry is a good model of knowledge and truth.

Well as you've pointed out the Greek word for maths wasn't math. Greeks studied mathematical problems by means of so called ruler and compass techniques. So the Pythagorean proof of the existence of an irrational number would involve geometrical diagrams. In short for Greeks of this period maths and geometry were the same thing.

Weather or not the reference is apocryphal the fact that Plato was a admirer of Pythagorus and his methods is well recorded and not, to the best of my knowledge, debated by any credible historian.

I freely admit the term philosophy is applied more loosely in the wider world but the philosophy you learn at university is very much based on the axiomatic approach of men such a Descartes and Plato. It can hardly be disputed that such an approach runs parallel to mathematic axiomatic logic and at least in the case of Greek philosophy was birthed from it.

This post has been edited by CodeWarrior on Aug 7 2005, 11:28 AM
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AllanO
Posted: Aug 8 2005, 12:53 PM
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QUOTE (CodeWarrior @ Aug 7 2005, 05:23 PM)
Well as you've pointed out the Greek word for maths wasn't math. Greeks studied mathematical problems by means of so called ruler and compass techniques. So the Pythagorean proof of the existence of an irrational number would involve geometrical diagrams. In short for Greeks of this period maths and geometry were the same thing.

I disagree I think basic arithmetic (addition) and counting would have been recognized as seperate disciplines. The key point is what we now think of math did not necessarily exist in Greek times and they did not necessarily have a word that is a good match to what we mean by mathemtics. What we currently mean by geometry may be somewhat narrower (and also broader) than what the Greeks meant but as translations go I unhesitatingly say that geometry is a lot more accurate. The actual reason I bought it up is that the variant quote/translation lead to a tangent on the origin and meaning of math, so I had to address it f.

QUOTE

I freely admit the term philosophy is applied more loosely in the wider world but the philosophy you learn at university is very much based on the axiomatic approach of men such a Descartes and Plato. It can hardly be disputed that such an approach runs parallel to mathematic axiomatic logic and at least in the case of Greek philosophy was birthed from it.


I am not sure I would describe continental philosophy as studied at university as axiomatic. Also, even the philosophy of the likes of Plato and Descartes veers off the axiomatic for a good part of the time. Since our records of pre-Socratic philosophy are scanty at best I would not be so sure about the origins of Greek philosophy. Obviously, all scholarly traditions have parallels in that they all value good reasoning and in this way will all parallel in as far as possible many aspects of mathematical reasoning (at least in so far as mathematical reasoning is good reasoning). Similiarities in form are less important than the content. All that being said there is clearly a lot of cross-polination between certain kinds of mathematics and certain kinds of philosophy (far more than between philosophy and other disciplines).
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CodeWarrior
Posted: Aug 9 2005, 09:48 AM
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QUOTE (AllanO @ Aug 8 2005, 06:53 PM)
I disagree I think basic arithmetic (addition) and counting would have been recognized as seperate disciplines.

Then I am in agreement with the Greeks. I also would not class arithmetic as mathematics. Abstract algebra's can and do exist that do not operate from basic arithmetic. In my view the man who has mastered arithmetic is no more a mathematician that the man who has memorised the complete works of Shakespeare is a play write. The numerologist can tell you 2+2=4 the mathematician can tell you why.

QUOTE
The key point is what we now think of math did not necessarily exist in Greek times and they did not necessarily have a word that is a good match to what we mean by mathemtics.


Once again it depends on your definition of maths. Go back 500 years and ask your self where is the group theory, the differentiation or the matrix algebra. That doesn't mean there was no maths. The Greeks may have lacked many of the areas of maths that have since been discovered but the geometry of Pythagorus and Archimedes certainly qualifies as the axiomatic reasoning of quantitative (geometrical) statements. Indeed go back a 100 years in America and you will see in educational circles the term geometry was often used to refer to higher maths.

QUOTE
What we currently mean by geometry may be somewhat narrower (and also broader) than what the Greeks meant but as translations go I unhesitatingly say that geometry is a lot more accurate.


If for the Greeks all mathematics was studied through geometrical forms and there understanding of other mathematical fields were birthed from geometry is it so strange to suggest that in the mind of Greek, particularly one fond of Pythagorus, the 2 were equivalent.

QUOTE
I am not sure I would describe continental philosophy as studied at university as axiomatic. Also, even the philosophy of the likes of Plato and Descartes veers off the axiomatic for a good part of the time.


perhaps I should say then "philosophy as my universities prospectus portrays it."

QUOTE
Obviously, all scholarly traditions have parallels in that they all value good reasoning and in this way will all parallel in as far as possible many aspects of mathematical reasoning (at least in so far as mathematical reasoning is good reasoning).


Mathematical logic not good reasoning? huh.gif Even the theories of probability used to measure statistical data (some times incorrectly applied in that field) are based on mathematical logic even if the methods of collecting the data and the conclusions draw from it are completely illogical. That is due to people applying pseudo logic such as "as people who celebrate more birthdays tend to live longer celebrating birthdays must be good for your health." True rigourous precise and even yes pedantic axiomatic logic may be frustrating to work with for some people but I challenge you to explain how it might possibly be not good reasoning?

This post has been edited by CodeWarrior on Aug 9 2005, 09:50 AM
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