Four years ago, I read a book on the history of mathematics up to 1970 or so. It was very interesting up until the end. The last few chapters, though, were on lattices. The author claimed that lattices were taking math by storm, and that soon lattices would be one of the central objects in mathematics, and many other such claims similar to those now made in reference to category theory.

This use of lattices is also related to the fact that, even if preorders and lattices appare everywhere, the theory of such structures is not so important: that theory is rarely used to prove things outside itself, one usually prefers other tools and other languages. An extreme analogy: try to explain why mathematics is important to a experienced person who lives without mathematics (except the level of money arithmetic); quite possibly, no changes in the style of life will result.

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while lattices occur everywhere in mathematics, lattice theory does not seem to be a very popular area of mathematics since for some reason mathematics are not too interested in lattices as algebraic structures.

I would not put lattices (in the sense of universal algebra) in a role that was "central" to all of mathematics. Nor would I do that with a Denjoy Integral, or distributions, generating functions, nor even sets nor categories. All of those are tools or ways of thinking that are frequently being adapted or related to various other parts of mathematics. The interplay of the various notions is a reason (whether conscious or unconscious varies among individuals) to study and develop mathematics, in my not so humble opinion.

When we go to Dedekind-MacNeille completions, realizing that many inequalities involving max and min are lattice operations in another guise, and applying lattice theory in various ways to number theory, partial differential equations, computational complexity, and seeing how such a concept is useful to one's mathematical sphere, then one can say (or not) "Lattices are central to MY mathematics."

Answer to Question 1. In "The Many Lives of Lattice Theory," Gian-Carlo Rota wrote of "Professor [I. M.] Gelfand's oft-repeated prediction that lattice theory will play a leading role in the mathematics of the twenty-first century." So you'd better get on board now!

Lattices appear in many branches of mathematics. For instance, if one has a certain kind of object such as a topological space or a group, then one obtains a lattice by taking the collection of all subobjects of that object. For example, the collection of all closed subspaces of a Banach space form a complete lattice, and the collection of all subgroups of a group form a complete lattice as well. Furthermore, if one has a notion of a congruence, then the collection of all congruences form a lattice. It is therefore safe to conclude that lattice occur all over the place in mathematics. That being said, while lattices occur everywhere in mathematics, lattice theory does not seem to be a very popular area of mathematics since for some reason mathematics are not too interested in lattices as algebraic structures.

I think that the problem with lattice theory is that, although lattices appear everywhere in mathematics, they usually appear as objects not as a category. In other words, the mapping that assignes lattices to objects of a category $\mathbf C$ is often not an object part of a functor $\mathbf{C}\to \mathbf{Lat}$, where by $\mathbf{Lat}$ I mean the category of lattices with $\vee,\wedge$-preserving maps as morphisms. Usually, we only have a functor $F:\mathbf{C}\to\mathbf{Pos}$, that has only lattices in its range.

Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 at the Sorbonne. The complete list of 23 problems was published later, in English translation in 1902 by Mary Frances Winston Newson in the Bulletin of the American Mathematical Society.[1] Earlier publications (in the original German) appeared in Archiv der Mathematik und Physik.[2]

There are two problems that are not only unresolved but may in fact be unresolvable by modern standards. The 6th problem concerns the axiomatization of physics, a goal that 20th-century developments seem to render both more remote and less important than in Hilbert's time. Also, the 4th problem concerns the foundations of geometry, in a manner that is now generally judged to be too vague to enable a definitive answer.

The other 21 problems have all received significant attention, and late into the 20th century work on these problems was still considered to be of the greatest importance. Paul Cohen received the Fields Medal in 1966 for his work on the first problem, and the negative solution of the tenth problem in 1970 by Yuri Matiyasevich (completing work by Julia Robinson, Hilary Putnam, and Martin Davis) generated similar acclaim. Aspects of these problems are still of great interest today.

Following Gottlob Frege and Bertrand Russell, Hilbert sought to define mathematics logically using the method of formal systems, i.e., finitistic proofs from an agreed-upon set of axioms.[4] One of the main goals of Hilbert's program was a finitistic proof of the consistency of the axioms of arithmetic: that is his second problem.[a]

Hilbert's tenth problem does not ask whether there exists an algorithm for deciding the solvability of Diophantine equations, but rather asks for the construction of such an algorithm: "to devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers". That this problem was solved by showing that there cannot be any such algorithm contradicted Hilbert's philosophy of mathematics.

In discussing his opinion that every mathematical problem should have a solution, Hilbert allows for the possibility that the solution could be a proof that the original problem is impossible.[d] He stated that the point is to know one way or the other what the solution is, and he believed that we always can know this, that in mathematics there is not any "ignorabimus" (statement whose truth can never be known).[e] It seems unclear whether he would have regarded the solution of the tenth problem as an instance of ignorabimus: what is proved not to exist is not the integer solution, but (in a certain sense) the ability to discern in a specific way whether a solution exists.

At least in the mainstream media, the de facto 21st century analogue of Hilbert's problems is the list of seven Millennium Prize Problems chosen during 2000 by the Clay Mathematics Institute. Unlike the Hilbert problems, where the primary award was the admiration of Hilbert in particular and mathematicians in general, each prize problem includes a million-dollar bounty. As with the Hilbert problems, one of the prize problems (the Poincar conjecture) was solved relatively soon after the problems were announced.

That leaves 8 (the Riemann hypothesis), 13 and 16[g] unresolved, and 4 and 23 as too vague to ever be described as solved. The withdrawn 24 would also be in this class. Number 6 is considered a problem in physics rather than in mathematics.

The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the patterns in nature, the field of astronomy and to record time and formulate calendars.

Many Greek and Arabic texts on mathematics were translated into Latin from the 12th century onward, leading to further development of mathematics in Medieval Europe. From ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation.[11] Beginning in Renaissance Italy in the 15th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day. This includes the groundbreaking work of both Isaac Newton and Gottfried Wilhelm Leibniz in the development of infinitesimal calculus during the course of the 17th century.

The Ishango bone, found near the headwaters of the Nile river (northeastern Congo), may be more than 20,000 years old and consists of a series of marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either a tally of the earliest known demonstration of sequences of prime numbers[13][failed verification] or a six-month lunar calendar.[14] Peter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division, which he dates to after 10,000 BC, with prime numbers probably not being understood until about 500 BC. He also writes that "no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10."[15] The Ishango bone, according to scholar Alexander Marshack, may have influenced the later development of mathematics in Egypt as, like some entries on the Ishango bone, Egyptian arithmetic also made use of multiplication by 2; this however, is disputed.[16]

Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (modern Iraq) from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity.[19] The majority of Babylonian mathematical work comes from two widely separated periods: The first few hundred years of the second millennium BC (Old Babylonian period), and the last few centuries of the first millennium BC (Seleucid period).[20] It is named Babylonian mathematics due to the central role of Babylon as a place of study. Later under the Arab Empire, Mesopotamia, especially Baghdad, once again became an important center of study for Islamic mathematics. 589ccfa754

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