Mathematical Logic is divided broadly into four areas — model theory, recursion theory also known as computability theory , proof theory and set theory — that have common origins in the foundations of mathematics, but now have very different perspectives. There is also a strong interface between Logic and computer science, including topics such as automated reasoning and program extraction. In its most basic form, Combinatorics is concerned with arrangement of discrete objects according to constraints.
Combinatorics studies discrete structures such as graphs also known as networks and hypergraphs. This research area includes, for instance, algebraic and probabilistic combinatorics, combinatorial optimisation and Ramsey theory. Although both areas are of a relatively small size, they continue to produce research of an international standard.
The UK mathematical Logic community is small but continues to deliver research of international quality.
Permutations and binomial coefficients
The UK has strong international expertise in three main areas of Logic - proof, model and set theory - with model theory highlighted as a particular strength. UK expertise includes those actively working in computer science and philosophy departments with close ties to mathematics. In particular, exploiting links to Number Theory, Combinatorics, Algebraic Geometry, Topology and Geometric Group Theory were highlighted as potential opportunities for further strengthening of ties to areas of pure mathematics.
Links beyond mathematics e. Evidence source This is a rapidly evolving field of mathematics with connections to many research areas e. The UK has a world-leading reputation in this area, with particular strengths in topics such as extremal, additive, enumerative and algebraic Combinatorics. The UK's strength in these has been rewarded by high-profile awards and funding from the European Research Council. Despite recent growth in the number of researchers working in this area, the interface between algorithms, combinatorial optimisation and Combinatorics remains under-represented in the UK compared to communities working on this in the US and the European Union.
UK expertise in model theory continues to be world-leading, while strength remains in proof and set theory, but computability theory expertise has declined over the previous Delivery Plan.
- The Combinatorial Foundations of Information Theory and the Probability Calculus;
- Field-flow fractionation: analysis of macromolecules and particles.
- Minds Eye;
- Studies in Foundations and Combinatorics (Advances in mathematics: Supplementary studies)?
- McGraw-Hills GMAT, 2010 Edition.
In Combinatorics, the UK's standing has significantly increased over the last decade with extremal, probabilistic, algebraic and enumerative Combinatorics being at the forefront of research in this area. Research in Combinatorics is difficult to identify explicitly due to its underpinning role across the pure mathematics research areas. Both Logic and Combinatorics are underpinning fundamental research areas and so play a key role in supporting ongoing research in other areas of the mathematical sciences and other disciplines such as Information and Communication Technologies ICT. But doing an honest job of teaching traditional axiomatic not naive set theory like ZF to high school students would also present challenges.
The kind of string diagram calculus I have in mind was foreshadowed in the existential graphs of C. Peirce, and there is a good amount of discussion about this kind of thing over at the n-Category Cafe. Some of this is explained in the text on topos theory by Mac Lane and Moerdijk, and some is spread over more technical literature.
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The question is actually a little bit tricky. The formal logical backdrop behind ZFC is full first-order logic which everyone feels they understand at least intuitively and subconsciously — if we assume that background, we can jump right in and start doing ZF. Developing the internal logic of a topos is a fascinating process to me, anyway , and one can gain a great deal of categorical insight into the structure of first-order logic as a result, as well as gain a great semantic expansion thinking here of sheaf semantics, for example in the way one can interpret set theory.
But: it takes work , of a nature which seems to put off many traditional set theorists, recalling for example the extremely acrimonious debates and ad hominem attacks that took place on the FOM list in the late nineties. Thanks, Todd. The way I see it as an observer and this is expressed by the programming language metaphors, is that category theory is a different type of foundation than set theory and logic.
Since the frontiers of set theory and of logic are now much further away from basic ZF axioms or the definition of first-order theories, the relevance of different axiomatic approaches to cutting-edge set theory or model theory is not clear. Yes, I agree that category should be considered foundational in that conceptual sense as well.
The general attitudes toward categorists or categorical research in my own country, the US, are also pretty bad: funding for this activity is virtually non-existent, and it is very, very hard for a young categorist as such to eke out a living. Most will give up.
Perhaps attitudes in Europe and in your own country are somewhat more enlightened. So I think the impression needs some adjustment. But it is impossible to tell in advance…. My own opinion on the matter, if it worth something, is that math has no formal foundation; Whenever we are using sets and functions, categorical notions are already there. And whenever we are using categories, set theoretic notions are already there present.
Either are merely different ways to look on the same mathematical world. Mathematics just had to mature, and only at the time of people like Weierstrass it has matured enough for non contradictory definitions to replace the old ones. My feeling about sets is similar — sets work. It works. If we feel uncomfortable we sometimes technical formalities like Grothednick universes and the like, but those ad hoc solution seem artificial and moreover are mostly immediately forgotten; they are rarely used in the actual mathematics in any practical way outside of questions that has clear set theoretical character.
For Calculus, he said, the problem was mostly theoretical. They could tell right from wrong.
Thus Calculus was free to evolve, and the theoretical foundations could come when time was right. So The question of foundations has actual, significant influence on our picture of the world of sets. In practice, though, among things that serve as foundational in mathematics are naive set theory whose language show everywhere , arithmetic and basic algebra, basic group theory and abstract algebra, basic combinatorics, and even basic category theory. On the contrary, those are very specialized, specific branches of mathematics. You are commenting using your WordPress. You are commenting using your Google account.
You are commenting using your Twitter account. You are commenting using your Facebook account. Hadwiger, H. Combinatorial Geometry in the Plane. Pach, J. Combinatorial Geometry.
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