Monday, November 21, 2022

Making Finitary Deductions About Infinities

That is my 5-word definition of Mathematics. It is what distinguishes Math from every other field.

Some say that Math is the study of numbers, or the use of symbolic notation. But Music uses symbolic notation, and numbers are used by all the hard and soft sciences.

None of the empirical sciences ever encounter infinities. Cosmologists may talk about the universe having infinite extent, but there is no reason to believe that, and we cannot observe that. We only observe finite quantities.

And the sciences never make a finitary deduction either. An experiment might convince us of some fact, but it is really just evidence that makes an outcome 99% likely, or something like that. The experiment has to be refined and redone to become more and more sure of it.

Math has infinities all over the place. This is obviously true about work on limits, but it is also true about elementary statements like the Pythagorean Theorem. There are infinitely many possible right triangles, and the theorem gives a formula about all of them.

Even with all the infinities, the proofs always use a finite set of steps from a finite number of axioms. The proofs about the infinities are always strictly finitary.

Here is the Wikipedia definition of Mathematics:

Mathematics (from Ancient Greek μάθημα; máthēma: 'knowledge, study, learning') is an area of knowledge that includes such topics as numbers (arithmetic and number theory),[2] formulas and related structures (algebra),[3] shapes and the spaces in which they are contained (geometry),[2] and quantities and their changes (calculus and analysis).[4][5][6] Most mathematical activity involves the use of pure reason to discover or prove the properties of abstract objects, which consist of either abstractions from nature or — in modern mathematics — entities that are stipulated with certain properties, called axioms. A mathematical proof consists of a succession of applications of some deductive rules to already known results, including previously proved theorems, axioms and (in case of abstraction from nature) some basic properties that are considered as true starting points of the theory under consideration.

Mathematics is used in science for modeling phenomena, which then allows predictions to be made from experimental laws. The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model. Inaccurate predictions, rather than being caused by incorrect mathematics, imply the need to change the mathematical model used.

Here is Britannica:
mathematics, the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter. Since the 17th century, mathematics has been an indispensable adjunct to the physical sciences and technology, and in more recent times it has assumed a similar role in the quantitative aspects of the life sciences.
Here are some dictionaries:
The abstract science of number, quantity, and space. -- Oxford

An abstract representational system studying numbers, shapes, structures, quantitative change and relationships between them. -- Wiktionary

The science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations. -- Merriam-Webster

The study of the measurement, relationships, and properties of quantities and sets, using numbers and symbols. --

The science that deals with the logic of shape, quantity and arrangement. -- Live science

The study of numbers, shapes, and space using reason and usually a special system of symbols and rules for organizing them. -- Cambridge

These are pretty good, but do not distinguish Math from science well. Yes, Math is an area of knowledge that includes numbers, but the mathematicians do proofs, with infinite numbers and finite arguments.


  1. Dear Roger,

    ``Making finitary deductions about infinities'' sounded like a great isolation of a very, very important point --- even a crucial point.

    It also spurred me to figure out a definition on my own, for myself... For the time being, my definition would go something like this:

    ``Mathematics is a science that deals with inventing concepts of methods that are useful in [and are invented for the purposes of] measuring sizes/quantities [of actually existing objects, their attributes, characteristics, actions, etc.].''

    Apart from the parenthetically noted elements, which need not appear in the actual definition, the following points too are to be understood :

    It's to be understood that sizes [and quantities] exist in the physical world, but not the concepts of mathematics.

    As a consequence, objects of the physical world and their actions (including the sizes with which they exist and act) are *discovered*; however, not just the methods but even *concepts* of mathematics are *invented*.

    It's also understood that the nature of actions that objects actually take in the world (including their interactions with each other) crucially depend on the sizes with which they (and their aspects) exist and act. Thus the importance of quantities in physics, and the need to invent mathematical methods, so as to increase the power of our reasoning.

    It's also understood that deductions can only begin after a base of premises is already had, and that such a base is only inductively derived --- even in maths.

    Finally, it's understood that concepts of maths only refer to the *abstract* methods that are invented only to allow making measurements, albeit *indirectly* (abstractly). In other words, *all* the *object*s (including structures) that mathematics uses arise only in reference to, and are in a 1:1 correspondence with, *methods* of calculations / algorithms --- nothing more. Yes, there is a 1:1 correspondence between mathematical operations and mathematical objects and vice-versa --- but the operations/algorithms/methods have a hierarchical precedence.

    Mathematics is important, extraordinarily important, because it allows us to increase the range of our reasoning: With maths, you can talk about quantities without having to concretely measure them in each instance that you talk about quantities.

    Happy Thanksgiving.


  2. You definition might be a good definition of Applied Mathematics.

    1. What element(s) of mathematics does it leave out?


  3. Your definition says Mathematics is a science. It is not about the study of Math, but about how Math is used by the sciences. I would call that Applied Math. Pure Math is not a science.

  4. Many years ago a colleague told me she was going to pursue graduate work in Math, specifically Number Theory. My naive response was "What else is there?" i.e. that covers all of Math.
    Maybe it wasn't so naive.