What is the most fundamental area of mathematics?

[Hippasus]

By fundamental, I mean, 'what is that upon which all other mathematics is based'. I used to consider it as a duel between geometry and algebra (with the presupposition of arithmetic by its side), but now think the question could be expanded a bit. Fundamentality, importance or significance, and fruitfulness might be overlapping notions like a Venn diagram. For someone, it might be a matter of 'what is it upon which mathematics can be (potentially) most fruitfully expanded'. For someone else, it might be a matter of 'what is the most significant area of mathematics for my experience as a physicist in a laboratory and work desk'. For someone else it might be a matter of 'what is the most important area for teaching math from preschool to grade 12'.

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[JMCx4]

Other ... Number Systems. All of the fields of mathematics listed in your poll stem from & are therefore dependent upon numbers. /thread

 

[Hippasus]

Other ... Number Systems. All of the fields of mathematics listed in your poll stem from & are therefore dependent upon numbers. /thread

Ah man, forgot about that one. Added that and differential equations.

I think one could make a case for calculus and the like since change and dynamics are such a big part of the physical universe. Alternatively: abstract algebra. What is mathematics without the study of a set and operation(s) (i.e. algebraic structures)? Alternatively: linear algebra. Something like a nexus for higher mathematics. From two-dimensionality to unlimited dimensionality (the writing of matrices on a sheet of paper to the n-dimensional representation capabilities therein). Also look at all the applications there. Alternatively: mathematical logic. Where would mathematics be without Euclid and the axiomatic method? Proof, set theory, metamathematics, computation, etc.

 

[JMCx4]

You're missing my point. The most fundamental area of mathematics is "number systems", aka, "numeral systems": the notation systems which express numbers. Without structured number systems, those digits or other symbols representing the numbers cannot be effectively used in any of the mathematical areas you have cited - including Number theory.

 

[Hippasus]

You're missing my point. The most fundamental area of mathematics is "number systems", aka, "numeral systems": the notation systems which express numbers. Without structured number systems, those digits or other symbols representing the numbers cannot be effectively used in any of the mathematical areas you have cited - including Number theory.

I would say you're talking about abstract algebra or mathematical logic then.

If a number system has division it is built into it, it is more complex than the notion of a group, which presupposes a set and one or more operations for a structure (with closure). One can't divide in a group. They're too primitive. If one can divide, one has a field (with, for instance, real numbers).

For the case of mathematical logic, one cannot count without the notion of set. Number systems presuppose counting. I thought number theory would cover this area.

Otherwise you might as well be talking about archeology or history, which are no more mathematics than philosophy or literature are.

 

[JMCx4]

So you didn't REALLY mean "fundamental" then ... "adj., serving as a basis supporting existence or determining essential structure or function."

 

[Hippasus]

So you didn't REALLY mean "fundamental" then ... "adj., serving as a basis supporting existence or determining essential structure or function."

Fundamental for science, focused upon material existence, and fundamental for abstract disciplines. Those may involve two different senses of the term. I'm not sure what "supporting existence" is for you, or where you're getting this definition from, but I was asking a metamathematical question in a sense. What area of mathematics, if any, can serve as a basis for the whole of the subject?

It could be one of number theory, mathematical logic, or abstract algebra, from my point of view at present.

 

[JMCx4]

Thanks for clarifying. And my definition came from Merriam-Webster. Too pedestrian, I suppose.

 

[MakeTheGoalsLarger]

Arithmetic

 

[Fourier]

I doubt that you would get a consensus on this one though none of them really would satisfy "that which all other topics are based on". (With that definition logic would be the closest.) There is also a lot of interplay and overlap between these disciplines.

In terms of real world uses linear algebra may well be the most significant of the subjects we currently see in say a typical university program even though Calculus has traditionally been the go to course.

These days the world could use people with a better understanding of basic statistics and probability. The pandemic clearly showed the danger of mass ignorance in these subjects.

 

[Fourier]

Fundamental for science, focused upon material existence, and fundamental for abstract disciplines. Those may involve two different senses of the term. I'm not sure what "supporting existence" is for you, or where you're getting this definition from, but I was asking a metamathematical question in a sense. What area of mathematics, if any, can serve as a basis for the whole of the subject?

It could be one of number theory, mathematical logic, or abstract algebra, from my point of view at present.

Lets look at Number Theory for example. Where would it be without both abstract algebra or complex analysis? On the other hand abstract algebra obviously has its origins on extending what we know about the underlying objects of arithmetic. So which one sits on the shoulders of the other???

 

[Hippasus]

Lets look at Number Theory for example. Where would it be without both abstract algebra or complex analysis? On the other hand abstract algebra obviously has its origins on extending what we know about the underlying objects of arithmetic. So which one sits on the shoulders of the other???

There is a sense in which what is most fundamental for mathematics is supposed to (a) capture primitive objects of discourse, such as integers (and a group operation such as modular addition), as well as (b) have the ability to translate the whole of the rest of mathematics (perhaps such as axiomatic set theory). In this sense, I think number theory rests on abstract algebra, even though, from a historically-inclined vantage point, it is the other way around: abstract algebra rests on number theory. I am more inclined towards the former direction, perhaps due in part to my current studies, but I could be swayed either way.

For abelian groups, the elements are indeed isomorphic to integers, for nonabelian groups, the elements can additionally be signified as shapes (like the dihedral group D3), but even these nonabelian groups can be translated into the language of numerical systems as they are, for instance, in the Sn (permutation) groups. For instance, S3 is isomorphic with D3. More generally, knowing what we know from the history and development of mathematics, it is hard to imagine a world where the language of algebra and one-dimensional lines of symbolic discourse, infused with numbers, do not take precedence over, say, continuous objects.

When looking at the fundamentality question, intuition may suggest that the continuum of real numbers should have some sort of fundamental status as objects of discourse. It may be a matter of circumstance and aesthetic choices. For example, someone who is a topologist or differential geometer may shy away from numbers altogether in favor of manifolds or some such objects, if such an inclination is even possible in the contemporary world.

 

[adsfan]

I voted for Stats, because you can use that in many different areas, from chemical formulations to predicting winners in college basketball to reliability testing.

When I was in high school, I wish that our Geometry teacher had taken us on a field trip to a billards hall to see some practical applications of angles.

 

[Hippasus]

I voted for Stats, because you can use that in many different areas, from chemical formulations to predicting winners in college basketball to reliability testing.

When I was in high school, I wish that our Geometry teacher had taken us on a field trip to a billards hall to see some practical applications of angles.

That's why we watch/play hockey. To finally figure out what angles mean.

 

[JMCx4]

That's why we watch/play hockey. To finally figure out what angles mean.

If you're makin' 5 bucks a goal, you've made the hockey-billiards connection.

 

[Fourier]

There is a sense in which what is most fundamental for mathematics is supposed to (a) capture primitive objects of discourse, such as integers (and a group operation such as modular addition), as well as (b) have the ability to translate the whole of the rest of mathematics (perhaps such as axiomatic set theory). In this sense, I think number theory rests on abstract algebra, even though, from a historically-inclined vantage point, it is the other way around: abstract algebra rests on number theory. I am more inclined towards the former direction, perhaps due in part to my current studies, but I could be swayed either way.

For abelian groups, the elements are indeed isomorphic to integers, for nonabelian groups, the elements can additionally be signified as shapes (like the dihedral group D3), but even these nonabelian groups can be translated into the language of numerical systems as they are, for instance, in the Sn (permutation) groups. For instance, S3 is isomorphic with D3. More generally, knowing what we know from the history and development of mathematics, it is hard to imagine a world where the language of algebra and one-dimensional lines of symbolic discourse, infused with numbers, do not take precedence over, say, continuous objects.

When looking at the fundamentality question, intuition may suggest that the continuum of real numbers should have some sort of fundamental status as objects of discourse. It may be a matter of circumstance and aesthetic choices. For example, someone who is a topologist or differential geometer may shy away from numbers altogether in favor of manifolds or some such objects, if such an inclination is even possible in the contemporary world.

I'll speak to your last point first. The notion of what we call an irrational number, at least in the broadest sense goes back millennia. But the idea of a formal presentation of the real numbers is not that old. The beginnings of a formal definition of the real numbers can perhaps be traded back to Stevin in the late 1500's. Stevin also introduced the idea of decimal representation. The term "real number" is accredited to Descarte in the 17th century. Without a formal definition in place mathematicians of the time became ever more comfortable with the idea of a real number representing a point on the number line.

Our formal understanding of the real numbers really came about in the 19th century. The first really rigorous attempt to define R was, not surprisingly, due to Weierstrass who effectively defined real numbers using infinite sums. Weierstrass went on to investigate continuous functions in a formal way. Dedekind saw Continuity as curious topic and in an attempt to better understand how it could be interpreted within logic, he came up with his notion of cuts that we still use today as one of the two main approaches to formally defining R, teh other being the equivalence classes of Cauchy sequences.

All this suggests that while the notion of a real number is fundamental to mathematics it would be hard to argue that mathematics rests on its shoulders.

By the way. My field is Abstract Harmonic Analysis. I study Banach and operator algebras that arise form locally compact groups. (Mostly non-abelian).

 

[sdf]

Wow great discussion by people who dont quite understand what mathematics actually is

 

[TheGreenTBer]

Wow great discussion by people who dont quite understand what mathematics actually is

Well, by that comment you've certainly convinced me you're the world's foremost expert on mathematics.

I bow to your glory.

 

[Fourier]

Wow great discussion by people who dont quite understand what mathematics actually is

I am genuinely curious as to what you believe it is then.