Does Mathematics Exist?

I just love maths 😀 Awesome articles 😀

Surprised to find an article dealing with the philosophy of mathematics on an Engineering forum. This topic is one of my favorite reading pastimes , though I hardly grasp what I read. A comprehensive list of the theories on the <a href="https://en.wikipedia.org/wiki/Ontology" target="_blank" rel="nofollow noopener noreferrer">Ontology</a> status of Mathematics has been given on Stanford Encyclopedia of Philosophy - an excellent source for philosophy related stuff.
<a href="https://plato.stanford.edu/entries/philosophy-mathematics/" target="_blank" rel="nofollow noopener noreferrer">Philosophy of Mathematics (Stanford Encyclopedia of Philosophy)</a>

It is interesting to note that professional mathematicians (not just engineers) hardly concern themselves with these questions. They are mostly engaged in pursuit of finding elegant proofs.Those that do interest themselves in these matters are philosophers of mathematics (OP rightly pointed out to <a href="https://en.wikipedia.org/wiki/Bertrand_Russell" target="_blank" rel="nofollow noopener noreferrer">Bertrand Russell</a> and <a href="https://en.wikipedia.org/wiki/A._N._Whitehead" target="_blank" rel="nofollow noopener noreferrer">A. N. Whitehead</a> as prime examples. Though in modern age <a href="https://en.wikipedia.org/wiki/Hilary_Putnam" target="_blank" rel="nofollow noopener noreferrer">Hilary Putnam</a> and his mentor <a href="https://en.wikipedia.org/wiki/Willard_Van_Orman_Quine" target="_blank" rel="nofollow noopener noreferrer">Willard Van Orman Quine</a> are considered influential).

So let's get down to the discussion :-

The difference between everyday objects and mathematical objects is that they are abstract in nature. We never really doubt the existence of a table, a chair , the mountains , the stars and - thanks to science - electrons and quarks or for that matter EM fields. All these entities are present in our physical universe.

But there is more to the world than these concrete entities. We as thinking animals definitely feel that there is something common between a pair of shoes, a pair of stars , a pair of electrons , a pair of hands, etc. We are apt to call this common entity as the number '2'. By observing many collections of 3 objects we start to understand the number '3'. Similarly we are able to extract other numbers conceptually by observing other collections of various objects. And it doesn't end here. We all are aware of a circle as the locus of a point equidistant from another. But how many of us have actually come across an ideal circle in real life (I challenge the reader to find me a perfect circle in this physical universe). What happens is we observe approximately circular shapes in real life and after observing sufficient number of these we get some inkling of a common property which these objects more or less possess , namely circularity.
Similar is the case with the concept of straight lines , triangles and so on.

So the debate among philosophers is not about the existence of these mathematical objects. I.e no one really doubts the existence of these mathematical objects. But rather how is this 'existence' different from the 'existence' of the common concrete objects. The common concrete objects such as stars , tables etc have space-time co-ordinates. But what about the number '1729'. Where and when is it located ? Does it make sense to ask such a question ?

Some philosophers say that these mathematical objects are located in a world which is different than ours in the sense that it is timeless and is spaceless. Such philosophers are called <a href="https://plato.stanford.edu/entries/philosophy-mathematics/#Pla" target="_blank" rel="nofollow noopener noreferrer">Philosophy of Mathematics (Stanford Encyclopedia of Philosophy)</a>. But this school of thought has to tackle the question as to how objects in our crude physical world have these properties which mirror this Platonic world. What exactly is the connection between the two. (Note - The ancient philosopher Plato also included more complex but abstract ideas such as beauty , justice , good , love in this world)

Then there are other philosophers known as <a href="https://plato.stanford.edu/entries/philosophy-mathematics/#Int" target="_blank" rel="nofollow noopener noreferrer">Philosophy of Mathematics (Stanford Encyclopedia of Philosophy)</a> who more or less argue that these mathematical objects are merely mental constructions of intelligent beings such as us. These people are faced with the question as to what the status of mathematics would be if the universe were lifeless. Opponents are bound to argue -'It just ain't in the mind'

There is also another school known as <a href="https://plato.stanford.edu/entries/philosophy-mathematics/#Log" target="_blank" rel="nofollow noopener noreferrer">Philosophy of Mathematics (Stanford Encyclopedia of Philosophy)</a> which believes mathematics can be reduced to simpler <a href="https://en.wikipedia.org/wiki/Rule_of_inference" target="_blank" rel="nofollow noopener noreferrer">Rule Of Inference</a>. But then these people have to answer the question as to the existential status of logical truths instead of mathematical ones. The problem merely shifts to another domain , albeit one whose rules are more obvious than that of mathematics. 😉 . There is the famous case of Bertrand Russell trying to prove that <a href="https://en.wikipedia.org/wiki/Principia_Mathematica#Quotations" target="_blank" rel="nofollow noopener noreferrer">Principia Mathematica Quotations</a> from rules of logic alone. It took him nearly 380 pages to do so , only to realize later that there were problems in it. The most obvious things are often the hardest to prove.

There are many more schools of thought in this field. Interested readers can find them in the Stanford link I gave in the beginning. The nature of the subject is such that it lies outside the scope of scientific verification.
And it will mostly never be settled by the human race (like most other philosophical puzzles).

P.S - I have made good use of linking feature of the forum. The blue-colored phrases will take you to web pages which will give detailed info on the phrases. Interested engineers can visit in free time.
Happy reading. 😴
Sorry for long post, simply couldn't resist writing in details as it is favorite topic of mine.

Arnav JoshiSurprised to find an article dealing with the philosophy of mathematics on an Engineering forum. This topic is one of my favorite reading past times , though I hardly grasp what I read.
P.S - I have made good use of linking feature of the forum. The blue-colored phrases will take you to web pages which will give detailed info on the phrases. Interested engineers can visit in free time.
Happy reading. 😴
Sorry for long post, simply couldn't resist writing in details as it is favorite topic of mine.

Towards the end of 1969 the house journal of UNESCO carried a sort of competition to establish communication with potential alien visits. What was given were a series of distinct symbols and the readers had to decipher them. I also took part in that. The argument was that if Humans themselves could not decipher the message, it was highly unlikely that any alien would.

Assuming that the message would have to be at least partly mathematics, it was not too difficult to track down the digits 0 through 9. Then the four basic operators. The approximately= sign was a bit of a bother, though could be deciphered based on the value 3.14 following that. Then there was the ratio of the diameters of earth to moon.

The discussion amongst some of us interested was whether the decimal system had universal (literally) applicability or if it was because humans had ten fingers. After all there were some systems 12 based and twenty based.

bioramaniThe discussion amongst some of us interested was whether the decimal system had universal (literally) applicability or if it was because humans had ten fingers. After all there were some systems 12 based and twenty based.

Very interesting. We are at an advantage now , thanks to computers we are accustomed to binary , hexadecimal and octal systems too. I think we have a love for the decimal simply because we have ten fingers as you say.

Though I think the binary system would be more universal than any other system. As we know it needs only two symbols. And a being would start to use it after it had learned to construct a number system using the concepts of 'presence' and 'absence' of a single entity. To illustrate , we humans take absence of current as '0' and presence of it as '1', roughly speaking. ( However we don't seem to have made use of the concept of absence/presence while coming across decimal system. Else we would have had a system with a base of eleven. Zero corresponding to absence of finger, one-ten corresponding to the ten fingers . 😐 It is interesting to note that Romans did not have 0).

In Real Analysis , mathematicians use another abstract concept to create the numbers , that of a set. A set as we know is defined on the concept of 'membership'. Each set is defined purely by what members it has.
Entire number system can be created using only the null set.
A set having no members is the null set - ϕ . This is used to denote zero.
Then the set containing the null set i.e {ϕ} is used to denote 1. ( {ϕ} is diffeent from ϕ)
Set containing the previous two sets i.e {ϕ,{ϕ}} is used to denote 2.
And so on. Thus we have constructed zero and the positive natural numbers simply out of the concept of 'membership'. I will provide links to constructions of other types of numbers :-
<a href="https://en.wikipedia.org/wiki/Integer#Construction" target="_blank" rel="nofollow noopener noreferrer">Integer Construction</a>
<a href="https://en.wikipedia.org/wiki/Rational_number#Formal_construction" target="_blank" rel="nofollow noopener noreferrer">Rational Number Formal Construction</a>
Construction of real numbers
Each construction makes use of the more basic concept constructed earlier. But point is just out of the null set we are able to construct conceptually the entire number line.

Yes i think that Math exists and every subject is drives for Mathematics.