Think back to school, where we are taught mathematical facts, or theorems, such as the Pythagorean theorem and the fundamental theorem of calculus. These facts are true because they can be deduced by chains of logical steps. This is the standard of truth in mathematics: a mathematical statement is considered true and is called a theorem if there is a reasoned proof that the statement is true. Thus a mathematical proof is intended to be a demonstration of the absolute certainty of the theorem being proved. However, a proof is a chain of logical reasoning, and chains have to start somewhere. Behind every proof is a collection of basic assumptions, called axioms, about how the mathematical world works. These basic assumptions are the focus of our research.
Mathematics became more intricate and more abstract throughout the 1800s, largely due to great advances in algebra, geometry, and real analysis (the theoretical basis of calculus). Disagreements concerning the validity of certain proofs arose, and the need for a unified foundations of mathematics became apparent. Early attempts to provide these foundations were plagued by contradictions, such as Russell's famous paradox. These failures precipitated the so-called "foundational crisis in mathematics" of the early 1900s. In response to the crisis, David Hilbert, the greatest mathematician of his day, proposed what is now called "Hilbert's program." Hilbert encouraged mathematicians to seek the ultimate axioms, from which all mathematical statements can be either proved true or refuted as false; for which all proofs can be verified mechanically (nowadays, we would say by a computer); which are free of contradictions; and, critically, which can be proved to be free of contradictions using the axioms themselves. Such axioms would provide the ideal foundations, as they would answer any conceivable mathematical question without fear of contradiction.
In the 1930s, Kurt Gödel surprised the mathematical world with his incompleteness theorems, which imply that there can be no single collection of axioms founding all of mathematics as Hilbert desired. Part of what Gödel showed is that reasonable collections of axioms cannot prove themselves to be free of contradictions. Thus there is no solid foundation for all of mathematics; we can never know for sure that there is no contradiction lurking among our basic assumptions. From the work of Gödel, Tarski, Turing, and others, we now know that axiomatic systems form a sort of tower. The bottom levels correspond to weak axioms, where few theorems can be proved but the foundational footing is strong. The upper levels correspond to powerful axioms that can prove many theorems, but whose foundations are much shakier.
In the 1970s, Harvey Friedman initiated a program called "reverse mathematics" whose goal is to pinpoint exactly how far up the axiomatic tower one needs to climb in order to prove core mathematical theorems. This is interesting because by exactly determining what axioms are required prove a certain theorem, we exactly determine the foundational commitment we make by accepting its proof. There are potential practical benefits to such an inquiry as well. Weak axioms tend to be algorithmic in nature, so if a theorem can be proved from weak axioms, then sometimes computational information can be extracted from the proof. Conversely, if a theorem requires strong axioms, then this can mean that no such extraction is possible.
In this project, we analyze key theorems from topology (the study of mathematical spaces) in the style of reverse mathematics. To date, this has only been done in a fairly piecemeal fashion, despite topology being central to modern mathematics. Part of the problem is that topology is extremely general, whereas reverse mathematics works best when restricting to specific sorts of mathematical objects. We work to expand reverse mathematics and to help give a full account of the foundations of topology.
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