Daniel Dennett (philosopher) calls mathematics a scientific system attempting to be internally consistent without any direct empirical basis or meaning. He is sadly wrong. Lawrence Krauss (physicist) calls mathematics a kind of philosophy - a tool for thinking about things and working out "what if" sorts of questions. He is slightly less wrong. Wikipedia and most pedagogical sources define Mathematics as a Formal Science, declaring that it is "not concerned with the validity of theories based on observations in the real world, but instead with the properties of formal systems based on definitions and rules." This is also completely wrong. Mathematics is supremely concerned with what statements are objectively, universally true.
Any number of statements about what Mathematics is or is not have been made: Mathematics is a language. Mathematics is a tool. Mathematics is a game. Mathematics is puzzles. Mathematics is an arbitrary construct. Mathematics is imperialist male-centric in-group signalling. Bla bla bla. All mostly misinformed rubbish. And not only offered by people who never use it or who know nothing about it beyond what they were unable to grasp in high school. People who use mathematics professionally, and even people who consider themselves professional mathematicians, usually have not really thought about what this thing is, why it works, and what it means.
So what is Mathematics really? Let's break it down. The syntax and notation of mathematics should be considered separately to the facts, discoveries, or conclusions of mathematics. The way we communicate and work with mathematical ideas is an invention that has been developed over hundreds of years. Few people for instance can today understand and work through the mathematical notations of early discoverers like Newton, Euler, Gauss, or Leibniz. Also, different cultures that discovered mathematical facts and principles independently often have mutually unintelligible ways of expressing those facts. Mathematical systems of notation therefore share characteristics of a language: arbitrary, relative to the culture that created it, and not universal or unique.
And then there's the ideas themselves. The content of mathematics (as divorced from the syntax) are discovered truths about quantities (numbers), the relationships between quantities and groups of quantities, and truths about operations on quantities and the transformation and manipulation of identifiable quantitative objects. These facts, being discoveries and not inventions, are universal. They are also absolute, and not (as many claim) merely relative to a set of assumed axioms. I assert that the basis of mathematics is not axiomatic, but empirical beginning with the empirical existence of integer counting numbers and their properties, and extending to the empirical, discoverable, and absolute properties of idealized geometric objects, spaces, operations, and functions. Once we understand and get over the problem of syntax, it is easier to understand mathematics as the following:
Mathematics is the taxonomy of discovered quantities, quantitative objects, their properties, and of mathematical operations, and the discovered relationships between quantities, operations, and quantitative objects. Mathematics makes use of arbitrary and non-unique invented systems of syntax and notation in order to document and explore this taxonomy.Mathematical quantities and quantitative objects exist not because we (or any other agency) call them into existence, but because of nothing more than the possibility of the existence of one, two three, or any other number of distinguishable things, be they asteroids, universes, atoms, oranges, or just abstract units or groups of units. And that is a very, very low bar for the existence of anything - so low that one may as well accept that mathematics is as self-existing as anything could be, existing and awaiting only discovery by competent observers.
Meanwhile the syntax and notation of mathematics are clearly human inventions. They are somewhat arbitrary in that respect, are not unique (i.e. by no means the only possible systems of notation), have regional dialects, and clearly evolve over time. While each system of syntax strives to be internally consistent and unambiguous, they are not always perfect. Syntax, like language, must be learned from other users of the syntax and through determined effort. We develop with effort and practice the facility to interpret and manipulate the syntax and notation of mathematics in order to read, manipulate, or write universal mathematical ideas.
Syntax and notation allows us to do three things.
1. Using syntax one can read and interpret a mathematical statement. This may be in the form of defining a stand-alone mathematical object (quantity or group of quantities) or in the form of a statement about a relationship between quantities, usually in the form of an equality relationship involving an operation. Mathematical statements often express information about how aggregate quantities are comprised of or related to other quantities, e.g. by addition or multiplication. The area of a rectangle is the quantity representing the base of the rectangle multiplied by the quantity representing the height. The length of an object in centimeters is related to the length of that object in inches through multiplication by 2.54. These cumbersome statements of fact are abbreviated succinctly using an invented notation: A = B*H; 1 in = 2.54 cm.
2. Through the use of a suitable syntax we can manipulate mathematical statements to find equivalent statements. The statement "This tree weighs five tonnes," stripped of the ambiguity and possible smarty-pants alternative meanings of a natural language statement, becomes Wtree = 5 t. As such it can be manipulated to express what else can be known from this statement alone. This includes the conclusions that two identical such trees would necessarily weigh 10 tonnes together, a tonne is one equal fifth part of this tree, dividing the tree into two parts of equal mass must yield parts of 2.5 t each, and so on. Notice that such manipulations and transformations do not add any new knowledge, but at best re-state the existing knowledge in order to be applicable or relevant to specific questions one may ask. It tells us only what we already know, although sometimes in a way that we did not at first appreciate.
For example, if we know that the speed of a certain body increases by 32 feet per second each second, it is pretty obvious that after 3 seconds of accelerating from rest, the object must be travelling at a speed of 3 x 32 = 96 feet/sec. This we know by mere extension or re-statement of the premise. No new information is required. But far less obvious is the fact that after 3 seconds it must also necessarily be 144 feet away. This non-obvious fact is not new information; it is actually contained within the premise. It only becomes obvious when the statement, expressed in mathematical syntax, undergoes valid syntactical transformations and manipulations leading to other entirely equivalent statements which we can then interpret. This is not generally possible with natural-language statements.
Mathematical manipulations of the syntactical expression of a statement accepted as true, done in such a way that each re-statement is also true, permit us to uncover many other true statements implied or required by the original one. It may be the case however that not all possible true statements about the premise can be discovered. The only guarantee is that if each manipulation is valid, the result is a true re-statement of the premise.
3. We can express mathematical ideas including asking questions about quantities and testing hypotheses about the relationships between mathematical objects. A mathematical object is sometimes literally an object like a line, a triangle, or a sphere, but more generally it is an identifiable collection of numbers, often which have a simple rule for determining which numbers belong to the object. Sometimes the numbers have to be in a specific order; sometimes it is something simple like "all numbers less than five."
For example, one may ask in natural language, "Which is the larger of the two - the area of a circle of diameter equal to one Glaaarrghtoot (aka a Glaaarrghsnaffian Inchmeter - 1/100,000 the mean circumference of Planet Glaaarrghsnaffia VII), or the area of a triangle, each side of which is also exactly one Glaaarrghtoot?" And while philosophers and theologians unconstrained by knowledge endlessly dispute the meaning of words like "circle," "diameter," "triangle," "planet," and "equal," and while engineers Space-Google the mean circumference of G. VII in Spaceyards, we can cut right to the chase using mathematical syntax:
√3 >? 𝜋
Mathematical discoveries include all numbers; groups of numbers; relationships between numbers; mathematical objects including geometric shapes, functions, and other identifiable groupings of quantities; operations that transform numbers, groups of numbers and objects; and relationships between mathematical objects.
These discoveries begin with the discovery of 1. The unit. A thing. Any single thing. Then along comes another thing, and we immediately discover 2. Two things. The idea of two. Two of something. Also, we make the discovery that one and one is two; or that two ones is two, two divided equally into two is one, and one removed from two is once again one. In our notation,
1 + 1 = 2
1 x 2 = 2
2 / 2 = 1
2 - 1 = 1
By careful observation of two units both together and apart, we have also thus discovered (not invented) the operations of addition, subtraction, multiplication, and division. Then we discover 3, 4, 5, and all the other cardinal numbers, and a myriad of facts about the relationships between them. We discover even numbers, odd numbers, square numbers, prime numbers, factors, divisors, etc in endless variety.
We can also discover without any further assumptions (or axioms) whatsoever the existence of an infinity of numbers between the cardinal (integer) numbers, as well as negative numbers. These non-obvious numbers are called into existence by the very existence of the operations we discovered at the start. Because Division empirically exists (you can divide objects or collections of things and count the results), non-integer numbers must therefore also exist. Five must ultimately be capable of being divided by two, for instance. Because Subtraction is a thing, negative numbers must therefore also be a thing. You can have an actual deficit of frogs - negative frogs - if someone owes you a frog.
Because an endless series of dividing whole numbers have no reason not to exist, irrational numbers (not expressible by any finite number of divisions of whole numbers) likewise are permitted to exist, and it can be shown that they do. Even less obviously, if multiplication is to be a logically self-consistent thing that exists, "imaginary" numbers must also exist - numbers which when multiplied together produce negative numbers, which we already know exist.
Besides quantities, there are endless other kinds of mathematical objects to be discovered: the point, the line, the plane, points on a plane, geometric shapes on the plane as identifiable groups of related points, n-dimensional spaces, and n-dimensional geometric objects. There are functions in endless variety: groups of numbers that are related to each other through a sequence of operations. y = 5x. y = sine(2x). All just awaiting discovery, and the discovery of their numerous obvious and not-so-obvious interrelationships. New techniques and better systems of syntax often need to be invented in order to more easily work with more complicated discovered objects.
But you may say all these "empirical" discoveries are merely of abstract ideals, not objects of physical existence. What is the connection then to the physical universe? Why do so many natural phenomena lend themselves so well to mathematical descriptions? What is the nature of the strange link between the real world and the purely abstract world of mathematics?
We need to walk back some of the question-begging smuggled in with these questions. Firstly, it is in no way "purely abstract" to observe that discrete items in the physical universe correspond to the cardinal numbers one discovers in mathematics. This is, indeed, how the cardinal numbers were originally discovered. One rock. Two rocks. Another makes three. One bear. Two bears. Holy shit - run for your lives! In no way is this purely abstract or hypothetical. One gallon - two gallons - not to mention the practically unlimited divisibility of gallons of liquids into smaller non-integer quantities. Mathematics is simply not the abstraction that so many have claimed it was.
Natural phenomena and the mathematics that describe them are likewise not the separate entities that the above questions presuppose, either. We discover natural phenomena at about the same time we discover the mathematics that describes them precisely because they are often one and the same. The inverse square law of physical phenomena such as gravity, radiation, luminescence, electric fields etc are not eerily mathematical due to some kind of conspiracy or fine tuning, or some deep mystery of surprising profundity, nor is the mathematics "just a model of reality." Rather, all these physical laws are nothing more than re-statements of the rather mundane mathematical truth discovery that the surface of any sphere increases as the square of its radius. Exponential radioactive decay and the law of half-lives is not atoms being artificially forced to obey an invented mathematical abstraction by some mysterious conspiracy; instead it is merely an instance of probability (another mathematical discovery) happening to large numbers of objects, not abstractly, but in real life.
It is not at all mysterious, nor should it be, that mathematics works so well in the natural and applied sciences, any more than it should be confusing that one rock and another rock makes . . . two rocks. It is a basic truth about things in the universe that they represent and are represented by numbers. Numbers have relationships that we can discover, and those relationships are again reflected in the real objects in the universe that embody these numbers, as a direct consequence of embodying those numbers. What's interesting is that while numbers are readily embodied by physical objects, they can also be embodied by abstractions. Numbers don't even need a universe in order to exist.
The historical development of mathematics is a confused story of simultaneously discovering mathematical objects while struggling to invent ways of talking about them. These are often ad hoc shortenings of natural language descriptions that evolved into some kind of operative syntax. This can easily account for why it is not obvious to more people - even mathematicians - that mathematics is really two very different things bound together: a taxonomy of discovered universal truths about quantities, and an invented arbitrary system of syntax needed to talk about them.