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In science, we have laws through which we can explain various phenomena. It seems all of it can be reduced to a few basic laws. This is the idea of reductionism. It is also possible that we cannot reduce everything to a few basic laws and that there are emergent laws. This is the idea of emergence.

Now, similarly in math, we have certain axioms through which we prove theorems and other results. Can all results be proved using certain axioms i.e. does math follow reductionism OR can there be emergent laws in math too which would mean that a set of axioms won't explain all mathematical results?

Hope this makes sense.

Shaun
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user422489
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    hint Godel's incompleteness theorems. –  Sep 01 '17 at 12:16
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    This is a very broad question, but you might want to read about Godel's Theorem regarding the intrinsic limitations of axiom systems. – lulu Sep 01 '17 at 12:16
  • @lulu here's the numberpile video ( okay the first one) : https://www.youtube.com/watch?v=O4ndIDcDSGc –  Sep 01 '17 at 12:17
  • @RoddyMacPhee thanks. – lulu Sep 01 '17 at 12:22
  • Great question and train of thought! – layman Sep 01 '17 at 12:22
  • https://math.stackexchange.com/questions/731640/can-mathematics-be-traced-back-to-a-fundamental-system-of-truths/731659#731659 – JobHunter69 Sep 01 '17 at 12:23
  • Are u basically asking that ^^^? – JobHunter69 Sep 01 '17 at 12:23
  • I also reasked a similar question but in the context of probability: https://math.stackexchange.com/questions/2098082/can-all-laws-be-derived-from-axioms – JobHunter69 Sep 01 '17 at 12:25
  • just some more answers one can take a look at. – JobHunter69 Sep 01 '17 at 12:26
  • Paradoxes can be encoded through axioms of mathematics too, and obviously you can't solve paradoxes (but you can recognize, and subsequently dismiss/avoid them). Does a set of all sets that don't contain themselves contain itself? – SF. Sep 01 '17 at 13:11
  • It seems to me that to any mathematician, the very idea of reductionism is completely preposterous. – Lubin Sep 01 '17 at 13:20
  • Thanks for the answers. I wanted to make an analogy with science. In science, the laws are not in our hands. We don't know how nature behaves. Maybe, there are finite laws or maybe infinite. But in math, we define all the rules. We know that these are the only laws possible. Then, how can there be things that can't be reduced to these laws? Seems unintuitive to me although still have to go through gödel's incomplete theorem to understand more. – user422489 Sep 01 '17 at 14:02
  • Related question or rather answer: https://www.quora.com/Do-G%C3%B6dels-incompleteness-theorems-restrict-the-scope-of-mathematics-and-thus-ultimately-of-physics – user422489 Sep 02 '17 at 09:28
  • Emergent laws can be reduced to basic laws. In fact, that's basically what emergent means -- that some observed law of physics arises not as a 'fundamental' law of the universe but merely as the product of more basic interactions. –  Sep 04 '17 at 21:26
  • Can you give some more details on what emergent law means? This is MSE and mathematicians may be less familiar with this than physicists. – Mikhail Katz Sep 05 '17 at 14:53
  • @MikhailKatz: Physics example: Ohm's law, as emergent from Maxwell Equations. The latter were formulated later, but are more fundamental; Ohm's law is their result and definitely not fundamental. In Mathematics, Pythagorean Theorem is emergent from the law of cosines, which is more fundamental. The law of cosines can be reduced to axioms. Yet the Pythagorean Theorem was found ("observed") first. Mathematics of any set of axioms could be represented as a tree, all theorems built either directly from these axioms or from "more fundamental" theorems (these are emergent). – SF. Oct 13 '17 at 16:34
  • I think of all mathematical statements in the universe of chosen axioms as a (directed) graph, which you can traverse as you wish and arrive from one statement to another. However, according to Kurt Gödel's proven results, regardless of the axiomatic system chosen (e.g. ZFC, ZFC+, etc), there are always nodes in this graph that are disconnected from the nodes containing the axioms. They exist in the universe but you can't reach them via any path from the axioms of your chosen system. – Aelx Dec 09 '21 at 11:14

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What you're probably searching for in a longer discussion is something along these lines. In short the answer is: There is no such thing yet and there will not be for a very long time. Given the fact that to build such a non-contradicting construct is currently beyond human comprehension. The reason I say that is due to the complexity of the entire field and very few span the entire cross section of application of pure and applied mathematics. The holy grail of pure mathematics is to find such beautiful things that tie all this together and make a coherent sense of the entire universe of mathematics. Roughly broken into the "four pillars of mathematics." Beyond this you have the ideas that there are many different kinds of set theories and they are created to work in different worlds so to speak, as such you'll find there are many algebras also for different reasoning. Most of the set theories fall painfully short of some sort of "latin of math language" if you will. Along with the fact that mathematics is actually created on the fly by new people every day to describe our world. There have been times that physics phenomena could not yet even be properly mechanically explained until the mathematics was invented to describe it. A person also has to contemplate mathematics as a way to take an un-exacting natural universe we live in and translate that to a fairly exacting set of language that mathematics really is.

Reference:

https://en.wikipedia.org/wiki/Foundations_of_mathematics

To give a taste of the depth and complexity (this is not all encompassing either.)

Foundations of Mathematics

Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory. The search for foundations of mathematics is also a central question of the philosophy of mathematics.

Philosophy of Mathematics

The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of mathematics in people’s lives. The logical and structural nature of mathematics itself makes this study both broad and unique among its philosophical counterparts.

Comments on Set Theory

The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment. The work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory. The work of analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics. Set theory is commonly used as a foundational system, although in some areas[which?] category theory is thought to be a preferred foundation.

Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venn diagrams. The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. This assumption gives rise to paradoxes, the simplest and best known of which are Russell's paradox and the Burali-Forti paradox. Axiomatic set theory was originally devised to rid set theory of such paradoxes.

Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as graphs, manifolds, rings, and vector spaces can all be defined as sets satisfying various (axiomatic) properties. Equivalence and order relations are ubiquitous in mathematics, and the theory of mathematical relations can be described in set theory.

Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica, it has been claimed that most or even all mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second order logic. For example, properties of the natural and real numbers can be derived within set theory, as each number system can be identified with a set of equivalence classes under a suitable equivalence relation whose field is some infinite set.

Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial; mathematicians accept that (in principle) theorems in these areas can be derived from the relevant definitions and the axioms of set theory. Few full derivations of complex mathematical theorems from set theory have been formally verified, however, because such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, Metamath, includes human-written, computer‐verified derivations of more than 12,000 theorems starting from ZFC set theory, first order logic and propositional logic.

Mathematics As Theory and Logical Reasoning

Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Since the pioneering work of Giuseppe Peano (1858-1932), David Hilbert (1862-1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. When those mathematical structures are good models of real phenomena, then mathematical reasoning often provides insight or predictions.

Through the use of abstraction and logical reasoning, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid’s Elements. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.

Perhaps Not

Galileo Galilei (1564-1642) said, ‘The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth’. Carl Friedrich Gauss (1777-1855) referred to mathematics as “the Queen of the Sciences”. Benjamin Peirce (1809-1880) called mathematics “the science that draws necessary conclusions”. David Hilbert said of mathematics: “We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise.” Albert Einstein (1879-1955) stated that “as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality”.

texasnick310
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