Define mathematics.What Is Ill Posed Philosophical Mathematics?

[chris_tams]

what Is Mathematics?

 

[matt grime]

Trawl through the philosophy section for plenty of debate on this timeworn, ill posed question.

 

[quartodeciman]

One can start with etymology. The Greek word "mathema" meant a study of something. The plural is "mathemata", or studies - deep investigations, maybe. One finds ancient writings and sermons "against mathematicians", which meant those who dabble in arcane mystical subjects. But the name seemed to be especially attached to reasonings about numbers and shapes by about the time of Plato and Aristotle.

It has become extremely difficult to say just what delimits and circumscribes mathematics any more; one can take the safe stance that mathematics is whatever people universally acknowledged to be mathematical practitioners say their subject is about.

I have known one person who actually bore an official title of Mathematician. That person had a US GS position with the US Navy as a civilian. The job actually consisted of writing computer algorithms.

 

[Gokul43201]

What exactly do you want to know ? Is there a specific question that you want answered ?

It's hard, in a brief statement, to describe "mathematics". It would have been easier a couple thousand years ago...but the field has grown from counting numbers and measuring figures to include several newer forms of "counting", "measuring" and "reasoning".

I'm certain even this non-answer will elicit the most terrible chastisement (such as "why do you distinguish counting from measuring ?"...i don't know) from the lightning bolt of matt grime... even though I've tried to save myself by lavishly using quotation marks which provide me with some kind of poetic license.

 

[Pattielli]

Here, I got it now: it is a science of numbers_ just the same as what people already mentioned though,
{^.^} {. . }{ . .}{*,*} :tongue2:

 

[Megus]

Maths is the most useful tool ever created

 

[Gokul43201]

Most useful...?? Hmmm...

 

[chris_tams]

yes youth!

 

[Gokul43201]

What about the knife ? Consider all that it gave rise to such as machining/cutting tools, razor blades, saws, blenders, choppers, scissors, lawn mowers, shears, wire cutters, diamond cutters, punches, meat cleavers, scalpels ? Now those are really useful...on the other hand, I know a few people that go through their lives blissfully unaware of the existence of such a thing as math.

Okay...this is debatable...

I should have gone for "language", i.e. the ability to elicit responses other than duh, unhhh, arrrrgh, ?, etc.

 

[killerinstinct]

a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement
www.cogsci.princeton.edu/cgi-bin/webwn[/URL]


The science of patterns and order and the study of measurement, properties, and the relationships of quantities; using numbers and symbols.
[PLAIN]www.iteawww.org/TAA/Glossary.htm[/URL]


The science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation ...[Britannica Online v 1.31, 1995]
[PLAIN]www.mathematicallycorrect.com/glossary.htm[/URL]


Genus: Science Differentia: The relationship and properties of quantities, through the use of numbers
[url]www.importanceofphilosophy.com/Dictionary.html[/url]


curriculum studies in the area of mathematics education; includes numeracy, arithmetic, geometry, statistics and algebra.
[PLAIN]www.standards.dfes.gov.uk/research/glossary[/URL]

Math is the language of Science, currently anyways (in my opinion)

 

[FlatlineLemon]

to paraphrase Eric Temple Bell "Mathematics is the beautiful language of numbers, when manipulated can paint amazing pictures"

 

[Icarus]

Kind of circular, defining math in terms of numbers, but how do you define numbers? within mathematics!

Besides which, many parts of mathematics are almost completely divorces from numbers. In my dissertation, the only numbers that appeared were a few low integers - and these were mostly used as indices.

My best effort is that mathematics is the study of what can be learned by Aristolean logic alone (other logics fit in this as they are studied with Aristolean logic). A mathematical theory starts with a set of definitions, undefined terms, and axioms, and builds on these foundations by proving theorems from the definitions and axioms. Thus Mathematics is the process that relates the theorems proved to the definitions and axioms with which one chooses to start.

Another definition: Mathematics is the study of similarity. I.e., what do diverse systems have in common?

A basic example of this: what do the fingers on your left hand have in common with the points on a star-shape? Well, you can match your fingers up one-to-one with the points and not have any fingers or points go unmatched. This is the basis of the idea of "number".

 

[Janitor]

mathematics is the study of what can be learned by Aristolean logic alone (other logics fit in this as they are studied with Aristolean logic). - Icarus

I come across the term "metamathematics" now and then. I gather it is the method(s) of reasoning about mathematics proper, rather than reasoning within mathematics. I wonder if there is always a clear distinction between mathematics and metamathematics, such that all mathematicians would agree whether a particular piece of reasoning belongs to one and not to the other.

 

[robert Ihnot]

It would be hard to always differenate between math and metamath, because Mathematical proofs such as the Godel's Incompleteness Theorem are structured in a mathematical way. The difference is more what is the intention of the proof, rather than the technique.

 

[Janitor]

Sounds about right. Thank you Robert.

 

[Icarus]

No - metamathematics is mathematics about mathematics. It works like this:

You have a mathematical theory - axioms, definitions, proofs. Unfortunately, you cannot create a powerful theory that is able to discuss itself - to use its mathematical tools in examining its own statements - in particular, to quantize its relations (such expressions as "for all relations X, ..."). If you try, you will find that this is inherently contradictory.

But in discussing a mathematical theory, such expressions are almost a necessity! How can we get out of this conundrum? By using a separate theory to discuss our original theory! I.e., we create a second mathematical theory whose variables are allowed to take on as values relations from the original theory. The second level theory is called "metamathematics". This sidesteps the paradoxes that would have arisen had we allowed such in the orginal theory, as metamathematical statements cannot reference themselves - only statements in the original theory.

Note that metamathematics is still mathematics, it is just that its subject of study is another mathematical theory.

 

[Janitor]

Thanks, Icarus. Have you worked your way through Godel's books?

 

[Icarus]

No, but I have gone through explanations of his proof. He essentially goes the other way from metamathematics: building a model for a mathematical theory within the theory itself. This is how he was able to create a statement which essentially said "I cannot be proved". The statement more accurately says "The same statement as me within the model cannot have a proof in the model". By an application of metamathematics, he shows that if the statement was provable in the theory, then so would it's model version be in the model, creating a contradiction.

 

[Dialog]

...ill posed question.

Define "ill".

Define "philosophy".