The history of mathematics proves the correctness of Russell’s statement."Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." -- Bertrand Russell, 3rd Earl Russell OM FRS, British philosopher and mathematician
For example, the ancient Greek mathematician Pythagoras, born in about 580 B.C., derived the Pythagorean equation, which is probably the most-often proved equation in math. It says that the sums of the squares of the sides of a right triangle are equal the square of the hypotenuse: a2 + b2 = c2. Without it, we would be unable to do things of little importance such as building houses.
Logic is the science of correct reasoning. What then is reasoning? According to Aristotle, reasoning is any argument in which certain assumptions or premises are laid down and then something other than these necessarily follows. Thus logic is the science of necessary inference. However, when logic is applied to specific subject matter, it is important to note that not all logical inference constitutes a scientifically valid demonstration. This is because a piece of formally correct reasoning is not scientifically valid unless it is based on a true and primary starting point. Furthermore, any decisions about what is true and primary do not pertain to logic but rather to the specific subject matter under consideration. In this way we limit the scope of logic, maintaining a sharp distinction between logic and the other sciences. All reasoning, both scientific and non-scientific, must take place within the logical framework, but it is only a framework, nothing more. This is what is meant by saying that logic is a formal science.
Mathematics is the science of quantity. Traditionally there were two branches of mathematics, arithmetic and geometry, dealing with two kinds of quantities: numbers and shapes. Modern mathematics is richer and deals with a wider variety of objects, but arithmetic and geometry are still of central importance.
Foundations of mathematics is the study of the most basic concepts and logical structure of mathematics, with an eye to the unity of human knowledge. Among the most basic mathematical concepts are: number, shape, set, function, algorithm, mathematical axiom, mathematical definition, mathematical proof.
There are three reasons for discussing mathematics in general philosophy:
Mathematics has always played a special role in scientific thought. The abstract nature of mathematical objects presents philosophical challenges that are unusual and unique.
Foundations of mathematics is a subject that has always exhibited an unusually high level of technical sophistication. For this reason, many thinkers have conjectured that foundations of mathematics can serve as a model or pattern for foundations of other sciences.
The philosophy of mathematics has served as a highly articulated test-bed where mathematicians and philosophers alike can explore how various general philosophical doctrines play out in a specific scientific context.
But mathematics is fallible, it is a human construct that has its origins in social and physical interaction. It is we who determine its axioms and its hypothetical/deductive logic. However, once its axioms and hypothetical/deductive system have been established then there exists an objective problem-situation. Like the game of chess, once the rules are made then consequences (a metaphor for theorems) follow. Mathematics is an autonomous objective practice that creates problems the solutions to which are independent of whether anyone has found a solution and independent of whether the academic community accepts the solution. Following Chalmers (1978) argument that science is a process without a subject, mathematics is a process without a subject because its development creates objective problem situations whether or not mathematicians realise the problem situation. A particular community may or may not practise a legitimate mathematics, and different social set-ups have created different contributions to a single (albeit multifarious) mathematics (e.g. Pascal's France and the China that created Pascal's triangle centuries before).
Mathematical theory does not reside in Popper's third world of ideas (the world of the objective content of thought), nor does it reside in Plato's world of forms; it is, instead, constantly produced and developed as a result of mathematical practice. The relation between consciousness and the objective, real, outside world is eloquently stated by Torrance:
If consciousness is an element of practice, what it reflects, or corresponds to, in reality, is always a specific problem-situation which practice confronts. As part of this reflection, it contains beliefs about the circumstances relevant to the problem. Such beliefs may reflect the circumstances more or less effectively, depending on their relevance, completeness, accuracy and so on. And these features depend, again, not only on the circumstances themselves, but also on the `position' or `angle' from which they are reflected, the condition of the reflecting instrument, and the purposes of the agent whose instrument it is. (Ellis Paul Torrance 1995)
The relations between origin and validity are much more complex here than in the case of the forms of the objective spirit.
Marx saw the problem clearly: `But the difficulty does not consist in realising that Greek art and epic are bound to certain social forms of development. The difficulty is that they still give us artistic pleasure and that, in a sense, they stand out as norms and models that cannot be equalled.' Just as it is clear that Copernican astronomy was true before Copernicus but had not been recognized as such. (John Lukacs, 1974)
Mathematics that has been developed for the pure pleasure of pure maths can prove to be uncannily useful in physics, sometimes a long time after it was first thought of.
General relativity asserts that massive objects curve the fabric of spacetime. To formulate it Einstein used geometric notions of curvature developed by Riemann in the 19th century.
A fascinating example is a particular geometric notion of curvature developed by the mathematician Bernhard Riemann in the 19th century. Riemann cared nothing about physics when he came up with his ideas, and he certainly did not predict the dramatic developments in physics that were to flow from Albert Einstein's pen at the beginning of the 20th century.
Yet, Riemann's ideas turned out to be just what Einstein needed to formulate his general theory of relativity. According to general relativity, the force of gravity is the result of massive objects bending the fabric of spacetime. To describe this bending Einstein needed to define the curvature of a geometric object without reference to a surrounding space the object is embedded in — and this is exactly what Riemann had done before him.
Another thing that sets the mathematics of physics apart from maths as applied in other sciences is its incredible precision. As one example of many, consider a number called the
It is also possible to calculate the value of
That’s an agreement to thirteen decimal places. In no other field of science do theory and experiment concord to such a spectacular extent.
Purely mathematical considerations continue to lead the way in modern physics, and they continue to prove impressively productive (see here to find out more).
Astrophysics depends on the absolute accuracy and the logical framework which mathematics offers.
Physics, as in the language of a physicist, is the study of matter and energy and to a layman it is simply the study of Nature. Astrophysics deals with the deeper insights and truths about the universe.
Mathematics combined with physics are two of the most basic and oldest among the sciences. From Archimedes to the 21st century, physics has developed to such dramatic heights that even the giant scientific philosophers of the past would be shocked to see its current discoveries, its uses and the giant computing machines created to probe its mysteries.
Thankfully, one need no longer carry those ubiquitous stacks of punch cards, bound together with thick rubber bands called a deck, around like we did in the late 60's, and stand in line to give the program decks to a person working behind a counter in the computer room where they'd be run. During hours of peak usage, it was not uncommon to wait in line for an hour or more. Then you'd wait for the print out to land in your box.
I happen to love physics and astrophysics, thanks to the encouragement offered me by my HS physics teacher, Mr. George Youngstrom. Had I not dreamed of German history and language since my youth as well, I might have taken a far different path into science and math in college.