Fundamental principles(?) in mathematics
As said already in one of my previous posts, David Goldberg and I had a nice discussion about “fundamental concepts” in mathematics. Our definition of “fundamental” was that,
- once seen you cannot imagine anymore having not known it beforehand and it completely changes your way of thinking and
- a somewhat realistic approach, i.e., when subtracted from the “world of thinking” something is missing.
So here is our preliminary list of things that we came up with – in random order – and a very brief, (totally biased) meta-description of what we mean by these terms. Of course, this list is highly subjective! For each of these “fundamental concepts” the idea is to have (about) 3 applications and try to distill the main core. There will be (probably) a separate blog post for each point on the list.
- Identity/Equality. Closely related to being isomorphic. The power of identity is so penetrating that I cannot even find a short explanation. Do I have to? (Aristotle’s first law of thought)
- Contradiction. Showing that something cannot be true as it leads to a contradiction or inconsistency. Closely related to this is the principium tertii exclusi or law of the excluded third as this is how we often use proofs by contradiction: a statement holds under various assumptions because its negation leads to a contradiction. If you do not believe in the law of the excluded thirdthen you obtain a different logic/mathematics. In particular, in these logics, usually every proof also constitutes some form of an algorithm as existence by mere contradiction when assuming non-existence is not allowed. (see also Aristotle’s second/third law of thought)
- Induction. Establishing a property by relying on the same property for smaller sub-objects.
- Recursion. Somewhat dual to induction: a larger object is defined as a function of smaller objects that have been subject to the same construction themselves.
- Fixpoint. The existence of a point that is invariant under a map. Equilibria in games.
- Symmetry. The notion of symmetry. Take a cube – rotating it does not really change the cube.
- Invariants. Think of the dimension of a vector space. Invariants are a powerful way to show that two things are not equal (or isomorphic).
- Limits. What would we do without limits? The idea of hypothetically continuing a process infinitely long. Think of the definition of a derivative.
- Diagonalization. One of my personal favorites. Constructing a member not being in a family by making sure it differs from all the members in the family at (at least) one position. Diagonalization often exploits self-references. An example is Cantor’s proof.
- Double counting. You count a family of objects in two different ways. Then the resulting “amounts” have to be identical. Typical example is the handshaking in graphs.
- Proof. The notion of proof is very fundamental. Once proven a statement remains true (provided constistency etc). Interestingly, it can be proven that some things cannot be proven. A good example for the latter is the existence of inaccesible cardinals which is consistent with ZFC.
- Randomness. Randomness is an extremely fundamental concept. One of my favorite applications is probably the probabilistic method. Think about Johnson’s –approximation algorithm for 3SAT or the PCP theorem.
- Algorithm. When considering a function we are often not just interested in what computes but in particular howit can be computed. In this sense the algorithmic paradigm is an additional layer to the somewhat descriptive layer of classical mathematics.
- Exponential growth. What we were particularly thinking about was the idea that a relative improvement bounded away from ensures exponentional progress. This is used regularly in different scaling algorithms such as barrier algorithms, potential reduction methods, and certain flow algorithms.
- Information. The idea that often a critical amount of informationis necessary to decide a property. Then fooling set like arguments can show that the information is not sufficient. Prime examples include the classical example that sorting via comparison needs at least comparisons, communication complexity, and query complexity.
- Function/Relation. Mapping one set to another. In particular important when the function/relation is homogeneous, i.e., when it preserves the structure.
- Density and approximation. The idea that a set (such as the reals) can be approximated arbitrarily well by an exponentially smaller set (such as the rationals). This exponential by polynomial approximation is also something that we are using in approximation algorithms, say, when we round the input. In this case the set of polytime solvable (rounded) instances is “dense” in the set of all instances. It can also be found in set theory when using prediction principles (such as Jensen’s diamond principle or Shelah’s Black Box) to predict functions on a stationary set by an exponentially smaller set.
- Implicit definitions. The concept of defining something not in an explicit manner but as a solutionto a set of contraints.
- Abstraction. The use of variables is so ingrained in us that we cannot even imagine to do serious mathematics without them. But abstraction is much more. It is the ability to see more clearly because we “abstract away” unnecessary details and we use “abstraction” to unify seemingly unrelated things.
- Existence (in the sense that Brouwer hated). One of the keywords here is probably non-constructivism and the probabilistc methods and indirect arguments are two promiment methods in this category. This was something that Brouwer despised: the idea to infer, e.g., existence of something merely because the contrary statement would lead to a contradiction (Brouwer’s school of thought denies the tertium non datur). The probabilistic method might have been fine with him. Although that is not clear at all as on a deep level we are merely trading an existential quantifier for a random one… long story…
- Duality. By duality we mean the wider idea of duality, i.e., for example the forall quantifier and the existential quantifier. Basically, when we talk about duality we often think about some structure describing the “space of positive statement” and a dual structure that describes the “space of negative statement”. In some sense duality is a form of a compact representation of the negation of a statement.
- Counting. Counting is again something that penetrates every mathematical theory. My favorite application of counting is the Pigeonhole principle.
- Hume’s principle (suggested by Hanno – see comments). Two quantities are the same if there exists a bijection between them. Somewhat related to “equality” however here we explicitly ask for the existence of a bijection. For example there are as many integers as their are rationals.
- Infinity (suggested by Hanno – see comments). The idea that something is not finite. With the notion of infinity I feel that the notion of countably infinite and uncountably infinite is closely connected. In fact the Continuum Hypothesis (CH) is such a case. It is consistent with ZFC and asserts that the first uncountably infinite cardinal is the size of the power set of the natural numbers (essentially the reals), i.e., whether ). However in other models of set theory is possible, by adding, e.g., Cohen reals.