B What is the building block of maths?

[Simon Peach]

TL;DR Summary

Sets or?

Sets are called the building block of maths, but why? To me sets are a collection of 'things'. Were as numbers are what give the 'things' a meaning. Myself I think that addition is the building block and maybe subtraction, all the other processes, addition and division are only quick ways of adding and subtracting, all the other processes in maths are only ways of shorting the addition or subtraction.

 

[andrewkirk]

We don't have numbers unless we can count.

What do we count? Typically things that have similar qualities. I'm serving in a shop with a bowl of mixed fruit on the counter. A customer comes in and asks for three apples. How do I avoid giving her two apples and a cash register, or one orange, one banana and one cleaning cloth? By concentrating on the set of apples.

I suggest that we can't meaningfully count, and hence can't have a concept of numbers, unless we first have a concept of a "set" - a group of things sharing a property, that we can decide to count.

BTW although sets may be the building block of maths, I don't see them as the base. Logic is where it all starts. Set theory is built from logic, and maths is built from a combination of set theory and logic.

 

[malawi_glenn]

Sets are called the building block of maths

Source?

Myself I think that addition is the building block

Addition is an operator, not an object in itself.

Were as numbers are what give the 'things' a meaning.

What is a number? There are several kinds of numbers.

Note that, say, the natural numbers can be constructed (recursively) using sets.
Let $0 = \lbrace \, \rbrace = \emptyset$
Then define for all $n \geq 0$: $n + 1 = n \cup \lbrace n \rbrace$
This will give you
$0 = \emptyset$
$1 = \lbrace \emptyset \rbrace$
$2 = \lbrace \emptyset ,\lbrace \emptyset \rbrace \rbrace$
$3 = \lbrace \emptyset ,\lbrace \emptyset \rbrace , \lbrace \emptyset ,\lbrace \emptyset \rbrace \rbrace \rbrace$
and so on.

 

[fresh_42]

To me, the building block of mathematics is the cipher $0$. Sure, counting was probably earlier, but it doesn't take much to compare a number of things with your fingers and toes. However, it was a magnificent moment in human history when someone decided to count something that isn't there! Marvellous!

The less romantic view is, that it was probably an accountant who needed a symbol for zero debts, some balanced sheet. Hence, a different way to look at it is accounting (Babylonians) and geometry (Greeks) because that's where it all started in our past.

 

[symbolipoint]

TL;DR Summary: Sets or?

Sets are called the building block of maths, but why? To me sets are a collection of 'things'. Were as numbers are what give the 'things' a meaning. Myself I think that addition is the building block and maybe subtraction, all the other processes, addition and division are only quick ways of adding and subtracting, all the other processes in maths are only ways of shorting the addition or subtraction.

You are on your way. You have some sense for some of the building blocks.

Somebody could explain "collections" and "Sets" to you better than I can.

A big part of Mathematics is "Number". You have been thinking about the fundamental nature of Addition and of Subtraction. A bit while later you may learn to focus on Addition and on Multiplication because of how these help in keeping with some rules or laws.

 

[fresh_42]

A big part of Mathematics is "Number".

And we still carry the geometric heritage of the ancient Greeks when we distinguish between rational and irrational numbers.

 

[jbriggs444]

In some ways, the logical structure of mathematics starting from some simple starting concepts or other is an elaborate fiction.

We were multiplying four and four to yield sixteen long before Russell and Whitehead, ZF set theory, the Von Neumann natural numbers or the Peano Axioms. We put foundations under our arithmetic after the fact.

It really does not matter. And we can even prove that it does not matter. Of course, those proofs involve set theory...

 

[fresh_42]

In some ways, the logical structure of mathematics starting from some simple starting concepts or other is an elaborate fiction.

Yes, but it sounds so much better than an accountant or tax officer, or the fact that it was slavery that enabled the Greeks to have time to think about geometry, and invent what we now call democracy.

What surprises me about logic, is that mathematics - in comparison to physics - more or less went over its crisis, shrugging its shoulders and continuing as usual. Russell's set of sets that doesn't contain itself as an element, the axiom of choice, Gödel's proof about unprovable theorems, the continuum hypothesis; none of them shook up mathematics in the long term, although each one is a strange riddle.

 

[symbolipoint]

Building Block?
??

The technology for writing. People had to find a way to show accounting onto something visual; drawing of shapes and relative or accurate locations would have become possible, too.

 

[mathwonk]

euclid (geometry), euler (algebra). at least that is where one might ideally begin to build a foundation.

 

[AndreasC]

BTW although sets may be the building block of maths, I don't see them as the base. Logic is where it all starts. Set theory is built from logic, and maths is built from a combination of set theory and logic.

Alas, the logicism program (trying to basically reduce math to logic) did not exactly succeed and was largely abandoned.

 

[Hornbein]

Alas, the logicism program (trying to basically reduce math to logic) did not exactly succeed and was largely abandoned.

You could be right but that isn't my impression. I thought that it succeeded but then there was nothing further to be done and the whole thing didn't yield anything new so everyone lost interest. I could be wrong though. Care to elaborate?

 

[Hornbein]

TL;DR Summary: Sets or?

Sets are called the building block of maths, but why? To me sets are a collection of 'things'. Were as numbers are what give the 'things' a meaning. Myself I think that addition is the building block and maybe subtraction, all the other processes, addition and division are only quick ways of adding and subtracting, all the other processes in maths are only ways of shorting the addition or subtraction.

As far as I know the idea was that numbers could be built starting from set theory but the reverse isn't attractive.

 

[Hornbein]

euclid (geometry), euler (algebra). at least that is where one might ideally begin to build a foundation.

Euclid was far ahead of his time and deserves his fame but he implicitly assumed things so it wasn't really foundational. It was Hilbert who axiomized geometry, but this never garnered mass appeal.

 

[Stephen Tashi]

TL;DR Summary: Sets or?

Sets are called the building block of maths, but why? To me sets are a collection of 'things'.

Myself I think that addition is the building block and maybe subtraction,

Yes the intuitive way to think of mathematical operations is to think of them like other processes we perform in life. We can think of them as something that takes time and effort - either by a human or a machine. However, this point of view hasn't been useful in doing mathematical proofs. For proofs, it is better to think of addtion as an operation defined by the set of all ordered triples of numbers such that the first two represent the numbers to be added and the third number represents the answer. So "addition" can be view as an ordered set of 3 numbers. A person who focuses on time and effort may object to conceptualizing addition this way on the grounds that nobody has the resources to actually generate this complete set of all ordered triples. However, to prove theorems about additions (i.e.about all possible additions) requires that we conceptualize a set of things without explicitly listing all their individual members.

If you study (or have studied) calculus, you will study the concept of limits. The intuitive way to think about a limit is that its is a number "approached" in a step-by-step way, as if a human or machine is exerting itself to get closer to the limit. However, if we consider the definition of limit that is actually used in proving theorems about limits, it contains nothing about such a step by step process and does not define the process of "approaching". This is my favorite example of the tension between the intuitive way of thinking about mathematical processes versus the way that is useful for proving theorems.

 

[martinbn]

Sets are called the building block of maths, but why?

This needs some context, and a lot was already said, but I will chime in too. I think it is because nowadays everything in mathematics is formulated using sets. "A group is a set with a binary operation such that...", "A topological space is a set with a collection of subset..." and so on. Everything is sets and maps between sets.

 

[martinbn]

This might be of interested http://www.ams.org/notices/201009/rtx100901121p.pdf

I've posted it before in a different subforum.

 

[mathwonk]

dear @Hornbein: You are correct of course, at least in a formal sense, but in my opinion, the popular view that Euclid is non - rigorous is somewhat over emphasized. Almost all of of it I myself consider extremely rigorous. Hartshorne, in Geometry: Euclid and beyond, does a really nice job of expositing the subject and incorporating Hilbert's improvements. But Hartshorne makes clear that he is only tweaking Euclid's presentation.

The minor degree of Euclid's carelessness is illustrated by an early example where he forgot to assume that if two quantities are equal, then half of each of them are also equal. Hilbert remedies this. But the main issue was the failure to clarify how a point divides a line, and a line divides a plane, into two "sides", although Euclid's language does make clear that he was assuming this. (Actually in my translation of Hilbert, from 1902, he attributes this study of ordering of points, and the plane separation axiom in particular to M.Pasch.) The other big assumption was that there exists a family of rigid motions of the plane, preserving congruence, but again Euclid's language makes clear he is assuming this. (Instead of assuming this, as you know, Hilbert chooses to make an axiom out of the SAS theorem that Euclid proved using this assumption.) My memory is not so good any more but I believe that is about it.

(Some people also feel that Euclid neglected to assume the Archimedean property when discussing similarity, but others are of the opinion he made this clear.)

Interestingly, Hilbert himself also made an error, in his Foundations, but in the other direction. In fact one of his supposedly independent axioms was superfluous, as was proved by E.H. Moore in 1902.
https://www.ams.org/journals/tran/1...-1902-1500592-8/S0002-9947-1902-1500592-8.pdf

Having always heard myself that Euclid was not rigorous, and hence avoided it, I was surprised to find I thought otherwise when I taught this material from Euclid, Hartshorne and Hilbert a few times late in my career. So I wish to reiterate my personal opinion that the material in Euclid is utterly foundational. The concepts there lay the foundation not only for geometry, but for real numbers, algebra (at least he solves quadratic equations geometrically), trigonometry (he proves the law of cosines), and calculus as well (through his use of limits to find volumes, although Archimedes of course goes much further). His work is also much more accessible to the beginner than that of Hilbert in my opinion.

To satisfy your logical reservations, I might recommend reading Euclid with Hartshorne in hand, and this is how I actually learned the subject. Still I think a novice will not notice any of the flaws in Euclid, and will benefit greatly from it as it is.

Perhaps our difference of opinion relates to my use of the word "foundational", or "building blocks", not as what would satisfy a logician, but what would benefit a beginner starting out to understand mathematics, and to identify material that underlies all later concepts.

pardon me for the lengthy post, whose points you know very well. peace.

 

[mathwonk]

in a way, the idea that sets are the building block of mathematics is perhaps a little like thinking that paper or canvas is the building block of art, rather than paint, or colors, or brushes.

i.e. a bald set is uninteresting. The interesting and substantive part is the relations, or correspondences between the sets. The elements of a group are uninteresting without the operations. it is the topology that is interesting about a top space, not the set the topology is defined on.

but without the set it is harder to define the interesting relations, just as it is hard to paint without canvas.

one might argue that equivalence relations are the building block of much of mathematics, but how to give an example without picking set to define one on?

If one wants to understand any sort of mathematical construct, after learning the definition of the structure it imposes on a set, one should immediately ask for the corresponding definition of isomorphism. I.e.; to understand any sort of mathematical object, you must know how to tell when two of them are essentially the "same".

In abstract language, one says "morphisms are more fundamental than objects". The first answer to the question linked next is excellent.
https://math.stackexchange.com/ques...s-are-more-important-than-objects-really-mean

 

[symbolipoint]

mathwonk, #19 and the reference, complicated. How do you tell a first-grader student what are the foundations or the building blocks of Mathematics? How to you go back maybe two thousand years and tell people then what are the building blocks of Mathematics? Can we remember that our first grade teacher helped us with instruction and exercise to learn about counting and recognizing even and odd numbers? If we do, we can easily(?) understand how the idea of Sets, which we would learn about in a few more years, would help us in understanding,..., things this way, ...

 

[mathwonk]

even in learning how to count, one is learning how to make correspondences betweens sets, such as the correspondence between objects on a table and fingers on a hand....my posts here are of course not aimed at first graders, but anything one understands, one can explain, in the appropriate terms, to young children. I have taught the book I of Euclid to a class of (bright) 8 -12 year olds.

edit: actually in teaching very young children, one faces unexpected challenges. It turned out my students had considerably more intellectual than physical dexterity, e.g. they could more readily imagine abstract geometric shapes than cut them out physically with scissors and cardboard. But when helped to do this, it did also help their visualization I think.

 

[AndreasC]

You could be right but that isn't my impression. I thought that it succeeded but then there was nothing further to be done and the whole thing didn't yield anything new so everyone lost interest. I could be wrong though. Care to elaborate?

Well, without getting into too much detail (since I can't right now), logicism was hoped to do a lot more than it actually could do, at least in the form pioneered by Frege and Dedekind. Especially after Godel's important discoveries, it was found that you simply can't possibly formalize the logic to prove every true thing in mathematics, and that you can't prove the consistency of your system within that system.

 

[Lemnisc8]

My two cents- sets are the "building block" of maths because they are irreducible. The whole of set theory just needs one extra-logical relation: membership. You can't break membership up into separate ideas. And all other branches of mathematics require at least two more concepts to work.

 

[krismartin121]

How do you tell a first-grader student what are the foundations or the building blocks of Mathematics? How to you go back maybe two thousand years and tell people then what are the building blocks of Mathematics? Can we remember that our first grade teacher helped us with instruction and exercise to learn about counting and recognizing even and odd numbers? If we do, we can easily(?) understand how the idea of Sets, which we would learn about in a few more years, would help us in understanding,..., things this way, ...

To explain to a first-grader what the foundations or building blocks of mathematics are, you can start by saying that mathematics is all about numbers, shapes, and patterns. Tell them that when we count things, we are using math, and when we add or subtract things, we are also using math.

Then, you can explain that the idea of sets is an important foundation of mathematics. A set is a collection of things that share a common characteristic. For example, a set of fruits might include apples, oranges, and bananas, because they are all types of fruit.

You can then go back in time and explain to people from two thousand years ago that the foundations of mathematics are the same as they are today. You can tell them that numbers, shapes, and patterns are still the building blocks of math. You can also explain that people have been using math for thousands of years to solve problems and make sense of the world around them.

It's also important to note that the way we teach and learn math has evolved over time. For example, our first-grade teachers helped us learn about counting and recognizing even and odd numbers, which laid the groundwork for more advanced mathematical concepts like sets. Today, there are many different tools and methods that teachers use to help students learn math, including games, manipulatives, and digital tools.

 

[symbolipoint]

@krismartin121,
That thinking is along the way mine is. What you composed was probably not very easy.
Like as I was thinking, most of us were taught to count, learn about place value, recognize the counting numbers as odd or even, in first grade. We were (likely, most of us) taught about Sets when we began Pre-Algebra or Algebra 1. This all helped, at least for me.

 

[Best Pokemon]

Yes, but it sounds so much better than an accountant or tax officer, or the fact that it was slavery that enabled the Greeks to have time to think about geometry, and invent what we now call democracy.

What surprises me about logic, is that mathematics - in comparison to physics - more or less went over its crisis, shrugging its shoulders and continuing as usual. Russell's set of sets that doesn't contain itself as an element, the axiom of choice, Gödel's proof about unprovable theorems, the continuum hypothesis; none of them shook up mathematics in the long term, although each one is a strange riddle.

I think you mean "unprovable tautologies" and not "unprovable theorems".

 

[fresh_42]

I think you mean "unprovable tautologies" and not "unprovable theorems".

I actually meant the existence of undecidable statements.