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What is Math?

Posted: Fri Apr 17, 2015 5:18 pm UTC
by jewish_scientist
Idea Channel made an interesting video about what math is. I am interested in what you think. What is math?

Is it the study of non-physical, but still existing, objects called numbers? If it is, do other non-physical, but still existing, objects exist? Could any non-physical, but still existing, object be alive or sentient entities?

Math is used to communicate information, so is it a language? If yes, then how does math imply information that the originator did not know about, such as the existence of Neptune? How would words not associated with mathematics (https://xkcd.com/55/) be 'translated'?

Is it an invention we made to understand the world? If yes, then what if people invented something that describes the world better than math? If no one is one a deserted island, does math exist there?

Re: What is Math?

Posted: Fri Apr 17, 2015 8:06 pm UTC
by Flumble
jewish_scientist wrote:Is it the study of non-physical, but still existing, objects called numbers? If it is, do other non-physical, but still existing, objects exist?

Given the way you used "non-physical and existing" in the former sentence: yes, basically all concepts and ideas. (I mention both words, because depending on your definitions one can encompass the other and vice versa –language one of those badly communicated concepts/ideas)

jewish_scientist wrote:Could any non-physical, but still existing, object be alive or sentient entities?

You should ask the philosophy department about this.
By the way, philosophy is one of the other sciences that studies non-physical things.

The word "math" is, as far as I know, mostly used as the study of geometry, arithmetic and algebra. It seems to be uncommon to think of logic and analysis when hearing "math". It's also common to call mathematical notation itself (that what is written down) "math".

jewish_scientist wrote:what if people invented something that describes the world better than math?

"better" means you want to compare two things, which requires logic, which is part of math.
Can you prove that anything better than math can exist? I'm not sure if there's a logical proof disproving that anything better than logic can exist, but any system that defies logic is bound to fail get your mom into my bed.
Is this metamathematics by the way? Because I'm not good at it.

Re: What is Math?

Posted: Sun Apr 19, 2015 12:04 am UTC
by doogly
I think this definition was originally proposed by Terry Tao:

0 - Math is Euclidean geometry and counting

1 - A mathematician studies math
2 - Math is whatever mathematicians introduce in their study of math

Repeat 1 and 2 recursively.

Re: What is Math?

Posted: Sun Apr 19, 2015 8:11 pm UTC
by Tirian
I believe you mean "iteratively", not "recursively", since you're going forward. :D

Having left academia and then returned to it, I define mathematics somewhat more holistically than most career academics I have met. I believe that mathematics is the study of systematic processes of decision-making. This is an entire iterative process of understanding the domain of the problem, deciding which factors influence that decision, collecting data, manipulating the quantitative data, applying the results of that calculation to reach a conclusion, and then reviewing the effectiveness of the conclusion and the process to make a more correct or more quick decision next time a similar situation occurs. IMNSHO, the fact that most math teachers feel that their work is entirely about the abstract manipulation of numbers and the fact that laymen feel that they don't use their math knowledge "in the real world" are inextricably linked.

What would happen if something "better" than math came along? According to my definition, it would be still be math. This happens all the time. Better models, better algorithms, and more sophisticated branches of mathematics arise regularly, and when people use them in place of the old processes for making decisions, they are still making decisions.

The remainder of your questions are deeply philosophical. Yes, concepts and ideas "exist" without having physical form. There are some who believe that these concepts exist outside the universe but within human perception (which is how every culture on earth independently realized that six things is more than four things no matter what language they use to describe that fact). Since the late nineteenth century, mathematicians have formally defined numbers according to the axioms of Peano and Dedekind and it can be proven that it isn't an accident that there's only essentially one way that it all could have worked out. Concepts are neither alive nor sentient by our conventional definitions of these terms. And it's impossible to conceive of, say, six and seven "deciding" to switch places on the number line, so I can't imagine of how they would manifest their sentience to us even if they did possess it.

Re: What is Math?

Posted: Mon Apr 20, 2015 1:53 am UTC
by Cleverbeans
Math is an imaginary game where you apply certain rules in combination to build new rules. Sometimes these rules are motivated by problems in the physical world, and sometimes we find parts of the world that match the rules we made up. Math can be applied to very specific cases, but as a general descriptive language for the world it fails miserably. Math is an entirely human construction, and without people there is no math.

Re: What is Math?

Posted: Tue Apr 21, 2015 1:40 pm UTC
by Paradoxica
Tirian wrote:I believe you mean "iteratively", not "recursively", since you're going forward. :D

Having left academia and then returned to it, I define mathematics somewhat more holistically than most career academics I have met. I believe that mathematics is the study of systematic processes of decision-making. This is an entire iterative process of understanding the domain of the problem, deciding which factors influence that decision, collecting data, manipulating the quantitative data, applying the results of that calculation to reach a conclusion, and then reviewing the effectiveness of the conclusion and the process to make a more correct or more quick decision next time a similar situation occurs. IMNSHO, the fact that most math teachers feel that their work is entirely about the abstract manipulation of numbers and the fact that laymen feel that they don't use their math knowledge "in the real world" are inextricably linked.

What would happen if something "better" than math came along? According to my definition, it would be still be math. This happens all the time. Better models, better algorithms, and more sophisticated branches of mathematics arise regularly, and when people use them in place of the old processes for making decisions, they are still making decisions.

The remainder of your questions are deeply philosophical. Yes, concepts and ideas "exist" without having physical form. There are some who believe that these concepts exist outside the universe but within human perception (which is how every culture on earth independently realized that six things is more than four things no matter what language they use to describe that fact). Since the late nineteenth century, mathematicians have formally defined numbers according to the axioms of Peano and Dedekind and it can be proven that it isn't an accident that there's only essentially one way that it all could have worked out. Concepts are neither alive nor sentient by our conventional definitions of these terms. And it's impossible to conceive of, say, six and seven "deciding" to switch places on the number line, so I can't imagine of how they would manifest their sentience to us even if they did possess it.


Have you ever taken a look at topology? Your pressing of the need for numbers fails to make a valid point, since size is meaningless in topology. The only thing that matters are relations between sets (compact spaces, n-dimensional manifolds, etc.)

Re: What is Math?

Posted: Tue Apr 21, 2015 8:21 pm UTC
by Tirian
I have studied topology, and I appreciate your point. That post probably deserved a few extra adjectives like "elementary math" and "high school math knowledge".

But topologies (and Lie algebras and the like) highlight my essential point of the holistic and iterative mathematical cycle. You have a real-world problem, you create an abstract model in which you can analyze the problem, you solve the problem in your model, interpret the solution in the real-world context, and then try that solution and twerk your model for next time if appropriate. There is certainly pure math in which we create and investigate mathematical structures without caring if they have a real-world application, but I still think it's beneficial to count the entire cycle as mathematical in nature.