Can mathematics be traced back to a fundamental system of truths?

Question

I'm not sure exactly how to state this question, or even if it belongs here. Still, I hope you will consider it, as I find it very interesting:

Most of the results I've seen in mathematics come from combining different concepts/definitions/etc in particularly pretty ways. If one were to trace the origins of mathematical thought, with what would it begin? That is, is it possible to trace mathematical thought back to a compact set of premises, from which all else follows more or less naturally once we can define it?

If anyone has any suggestions on topics or books (preferably accessible to undergraduates) I can look at to better investigate this topic, I would really appreciate it.

Answer 1

The important part is that there can never be one set of axioms to prove everything. Try googleing the incompleteness theorem.

Answer 2

This hugely depends on what kind of answer you want, because there are severe abstractions of the foundations of mathematics.

Roughly, we need a language and rules for how to use it (what kind of statements "make sense"), plus rules of inference. Then we also need some true statements to start with.

Over time all these components have evolved. If you read the Dover Edition of Euclid (translated and commented by Heath), you will find that it does not entirely stand up to modern scrutiny, and you will also find a lot of mathematical content hidden in the definitions. The language carries with it assumptions about mathematical content.

You could investigate the Peano Postulates for Natural numbers leading to arithmetic, or "ZFC" - Zermelo-Fraenkel Set Theory with the Axiom of Choice. These are widely adopted, but there are still foundational questions they don't answer, and other systems are available. Kurt Gödel's famous theorems essentially tell us that there is no perfect formal theory of mathematics - we will always have to make choices, and apparently important (and once apparently obvious to most) issues like the continuum hypothesis turn out to be independent of formal frameworks which people thought would prove them. Just like Euclid's Parallel Postulate in a previous generation.

Perhaps you could give a fuller description of your interest, because there are hundreds of books and resources, at almost as many different levels of mathematical sophistication. Douglas Hofstadter's "Gödel, Esher, Bach" is not perfect, but gives some good insight into some of the questions.

Answer 3

If one were to trace the origins of mathematical thought, with what would it begin?

Check out Frege's Theorem (it's not actually theorem, but rather a 'process') which he developed in Begriffsschrift and Die Grundlagen der Arithmetik.

Frege tried to get arithmetic (natural numbers and the usual properties between them) from a logical setting and in a formal and logical way. In doing this he formalized predicate calculus and syntax.

Frege assumed only a few laws (among them the infamous law V), the concepts of concept (otherwise known as predicate), extension and also the existence of a function that mapped each concept to its extension.

This was probably the most serious attempt to reduce mathematics to its bare essentials before Cantor's set theory. Unfortunately it was inconsistent due to the Russel's paradox.

It should be noted that Frege was against formalizing geometry in this setting. Frege sided with Kant in saying that geometry was analytic a priori and a model of geometry in his theory wouldn't really be geometry. Hilbert and Frege 'fought' about this. Hilbert eventually won. See The Frege-Hilbert Controversy.

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Is it possible to trace mathematical thought back to a compact set of premises, from which all else follows more or less naturally once we can define it?

Yes, it is. There are several ways to formalize mathematics: type theory, category theory, set theory...

Almost all mathematicians work under a foundational system in set theory named $\sf ZFC$.

Answer 4

Generally the compact set of premises isn't the starting point. It's something that is produced as part of the organisation of our knowledge. When we try to understand how the things we know relate to one another, how certain things logically follow from other things, what are the things we base our arguments upon, we find that there are some things that we cannot really explain to ourselves in terms of anything more simple or more obvious. We can only identify and collect these things, and show how everything else we know is derived from them.

But the resultant organisation of our knowledge as being wholly derived from these few fundamental intuitions shouldn't be mistaken for the historical process of how our knowledge actually came to be. A mathematical proof is not a record of either how the idea of a theorem formed, or the process by which the proof was found, and the body of organised mathematical theory does not represent the process by which mathematics is created.

The real basis of mathematics is our natural informal intuitions of number, distance, area, volume, direction, rhythm, more than/less than/the same, inclusion/exclusion, adding/taking away, multiplicity, division, shared/unshared properties, argument, straightness, inside/outside, etc, etc, etc.

Mathematics: The Loss of Certainty is a good book to read about the history of trying to set the foundations of mathematics. Suitable for undergraduates too.

Answer 5

Mathematics is at its core logical deductions from premises. There is no universal set of premises. You can still ask questions that ultimately must lead to the addition of another premise to answer other questions. The incompleteness theorem is an interesting piece of logical thought itself: http://rationalwiki.org/wiki/Essay:G%C3%B6del's_incompleteness_theorem_simply_explained

It is interesting and on-going work to try and derive minimal collections of premises that still logically lead to commonly used mathematics. But, we already know that there is not a universal set.

Mathematics also proceeds along other lines as soon as some collection of premises is interesting to consider to enough people to encourage someone to pursue it. Sometimes, the conjectures, theorems, and corollaries are even applicable to other people's problems.

Mathematics is nothing more and nothing less than logical deductions from premises. Some people are much better at logical reasoning than others. Transmitting that logical reasoning from the best minds across generations and across cultures gives us access to the incredible ability to produce all the wonderful technology and progress you see around you.

Mathematics lets us apply the brains of thousands of people on the logic problems that we encounter if we accept the premises that the logical deductions proceed from. The number of premises we could accept is infinite, and therefore the logical deductions we may wish to understand and store for future generations is limitless.

Some of these deductions that mathematics produces will only be useful as a game or diversion --- beautiful and interesting but basically "art". Some of these deductions will be used for incredible culture changing understandings and devices in Science and Engineering. It is not possible to tell the difference before hand. So, mathematics progresses along whatever lines its practitioners decide --- and a casual observer might hope for either 1) a strictly applied approach (where problems are presented to Mathematics from other disciplines) or 2) a thoroughly random perusal of premise space so as to get the most logical thought possible.