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What is mathematics?

Alexmahone

Active member
Jan 26, 2012
268
I'm a math major and therefore need to be able to define "mathematics" in a sentence. A few years ago, I would've said "the study of numbers", but with topics like "Set Theory" that have little to do with numbers, that definition seems inadequate.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,780
I'm a math major and therefore need to be able to define "mathematics" in a sentence. A few years ago, I would've said "the study of numbers", but with topics like "Set Theory" that have little to do with numbers, that definition seems inadequate.
Speaking for myself, I would say that mathematics is axiomatic theory.
That is, it defines arbitrary axioms, and builds theories based on those axioms.
This is what sets math apart from all other sciences at a lonely distance.
All other sciences are based on empirical observations and merely use math as a tool to describe them.

For instance, we have the axioms of rings of integers and the axioms of fields of real numbers. Theorems based on integers respectively real numbers follow from them. Similarly we have axioms of set theory, and theorems on set theory follow from them.

According to Mathematics on wikipedia there is no generally accepted definition. However, it does mention that:
Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions.
 

Alexmahone

Active member
Jan 26, 2012
268
Speaking for myself, I would say that mathematics is axiomatic theory.
That is, it defines arbitrary axioms, and builds theories based on those axioms.
This is what sets math apart from all other sciences at a lonely distance.
All other sciences are based on empirical observations and merely use math as a tool to describe them.

For instance, we have the axioms of rings of integers and the axioms of fields of real numbers. The theories based on integers respectively real numbers follow from them. Similarly we have axioms of set theory, and the theories on set theory follow from them.

According to Mathematics on wikipedia there is no generally accepted definition. However, it does mention that:
Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions.
Thanks a lot!

But how are the axioms defined? Are they motivated by reality? By language?
If the axioms are completely arbitrary, one could essentially build infinitely many mathematical theories. So, how do mathematicians decide which axiomatic theories to study?
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,780
Thanks a lot!

But how are the axioms defined? Are they motivated by reality? By language?
If the axioms are completely arbitrary, one could essentially build infinitely many mathematical theories. So, how do mathematicians decide which axiomatic theories to study?
Historically math started as a tool to help the other sciences.
After all, numbers already existed to count sheep before they were abstracted into an axiomatic system.
And lines, distances, and angles existed before they were abstracted into Euclid's Elements.
So it's quite common that common sense or an existing science is a driver to develop mathematical axioms and theories to assist them.
However, it's also common that some mathematician comes up with abstract axioms and theories, that later on are turned into practical applications. This is true for many algorithms in computer science that existed before computers were invented.

As wiki states it:
Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics, or mathematics for its own sake, without having any application in mind. Practical applications for what began as pure mathematics are often discovered.
 

Deveno

Well-known member
MHB Math Scholar
Feb 15, 2012
1,967
In my mind, mathematics is the study of logical structures divorced from context. Number is one of those structures.
 

Country Boy

Well-known member
MHB Math Helper
Jan 30, 2018
469
Thanks a lot!

But how are the axioms defined? Are they motivated by reality? By language?
If the axioms are completely arbitrary, one could essentially build infinitely many mathematical theories. So, how do mathematicians decide which axiomatic theories to study?
Yes, there are infinitely many possible mathematical theories. Sometimes mathematicians study specific theories because of their applications, sometimes because they are particularly elelgant.
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,193
My definition: Mathematics is the art of recognizing numerical patterns in the universe.
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,707
The definition in Webster's dictionary is
mathematics a science that deals with the relationship and symbolism of numbers and magnitudes and that incudes quantitative operations and the solution of quantitative problems

The Oxford English Dictionary says
the abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations
 

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
I get this question a lot. By especially my non-mathematician friends. And I usually respond with the following question as my answer.

"What is music?"

My point is that everyone can recognize what music is (well, not everyone, technically, but that's not important) but it is difficult to define.

Further, in my opinion, nothing is to be gained by such a definition. In fact, any definition of music taken seriously will only end up constraining the composer to conform to the definition. This is true for any art form.

Mathematics above all else is an art form, and I believe that it is best left undefined. One recognizes mathematics when one sees it and that should be enough.

But, if one is really looking for a formal definition then the best I can think of is that mathematics is the system of results occurring as logical conclusions of ZFC and underlying definitions (something in me died while writing this last sentence).
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,780
I get this question a lot. By especially my non-mathematician friends. And I usually respond with the following question as my answer.
In such cases what my non-mathematical friends get, is that I'm good at arithmetic.
And that I can help people with arithmetic and fractions.
Math is supposed to be something beyond that, but what that is remains vague.
To be honest, I wouldn't know how to explain it better to those friends other than that in math we tend to use letters instead of numbers.

In my current job I'm active at computer programming.
It's actually after I had already finished university in computer science that it became clear to me that most people have no knowledge or understanding of the word 'programming' or 'computer language' at all.
In other words, I can't explain my job to any non-programmer either.

Mathematics above all else is an art form, and I believe that it is best left undefined. One recognizes mathematics when one sees it and that should be enough.
Isn't saying that math is an art form already a kind of definition that limits its scope?
I'm fairly sure that not all mathematicians will see it like that.

But, if one is really looking for a formal definition then the best I can think of is that mathematics is the system of results occurring as logical conclusions of ZFC and underlying definitions (something in me died while writing this last sentence).
It's difficult for me to accept that math would be defined as something related to something named ZFC that I haven't even heard of. (Crying)
 

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
In such cases what my non-mathematical friends get, is that I'm good at arithmetic.
And that I can help people with arithmetic and fractions.
Math is supposed to be something beyond that, but what that is remains vague.
To be honest, I wouldn't know how to explain it better to those friends other than that in math we tend to use letters instead of numbers.
That is true. Those who haven't been exposed to mathematics tend to think that math is about multiplying big numbers in the head as quickly as possible.
I try to show them that is more than that by sharing some puzzles. I usually fire with the handshake lemma and the 6-people-in-a-party puzzle.

Isn't saying that math is an art form already a kind of definition that limits its scope?
I'm fairly sure that not all mathematicians will see it like that.
I do not think so. Art is not limiting. It only grows in its scope as time progresses. However, people do become prejudiced over time. For example, the old masters of music in India, who have been trained in classical Indian music, may not recognize the modern forms of music as music. Similarly, many mathematicians do not regard combinatorics as serious mathematics. They argue that it is just a bunch of tricks. But I'm sure this will go away with time.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,780
I do not think so. Art is not limiting. It only grows in its scope as time progresses.
Art does not usually have a practical application does it?
Instead it's supposed be aesthetically pleasing (or displeasing) to the human senses.
However, math does have practical applications. Even if only to assist other sciences (and art forms like music).
 

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
Art does not usually have a practical application does it?
Instead it's supposed be aesthetically pleasing (or displeasing) to the human senses.
However, math does have practical applications. Even if only to assist other sciences (and art forms like music).
Arts are not pursued in service of a practical application. But if an application is found, it's more of a serendipity. Mathematics, of course, has a lot of applications, but my feeling is that it is not developed with any application in mind. There can be a time lag of 100 years between the development of a certain concept and its application to the sciences.
 

MountEvariste

Well-known member
Jun 29, 2017
85
In my opinion think the subject most similar to mathematics is philosophy.

Perhaps mathematics is an art in the same vein that philosophy is an art?
 

RahulSharma123

New member
Nov 2, 2018
1
Mathematics is the science of Numbers, Quantity and concepts of everything. The Mathematics Is the essential term for the measurements. The important topics for the Mathematics is as follows
Algebra
Analysis
Combinatorics
Geometry and Topology
Probability and statistics
Computational Sciences