John Tabak - Collected Quotes

"A crucial difference between topology and geometry lies in the set of allowable transformations. In topology, the set of allowable transformations is much larger and conceptu￾ally much richer than is the set of Euclidean transformations. All Euclidean transformations are topological transformations, but most topological transformations are not Euclidean. Similarly, the sets of transformations that define other geometries are also topological transformations, but many topological transformations have no counterpart in these geometries. It is in this sense that topology is a generalization of geometry." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"A topological property is, therefore, any property that is preserved under the set of all homeomorphisms. […] Homeomorphisms generally fail to preserve distances between points, and they may even fail to preserve shapes." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"Abstract formulations of simply stated concrete ideas are often the result of efforts to create idealized models of complex systems. The models are 'idealized' in the sense that they retain only the most fundamental properties of the original systems. The vocabulary is chosen to be as inclusive as possible so that research into the model reveals facts about a wide variety of similar systems. Unfortunately, it is often the case that over time the connection between a model and the systems on which it was based is lost, and the interested reader is faced with something that looks as if it were created to be deliberately complicated - deliberately confusing - but the original intention was just the opposite. Often, the model was devised to be simpler and more transparent than any of the systems on which it was based." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"Although topology grew out of geometry - at least in the sense that it was initially concerned with sets of geometric points - it quickly evolved to include the study of sets for which no geometric representation is possible. This does not mean that topological results do not apply to geometric objects. They do. Instead, it means that topological results apply to a very wide class of mathematical objects, only some of which have a geometric interpretation." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"At first glance, sets are about as primitive a concept as can be imagined. The concept of a set, which is, after all, a collection of objects, might not appear to be a rich enough idea to support modern mathematics, but just the opposite proved to be true. The more that mathematicians studied sets, the more astonished they were at what they discovered, and astonished is the right word. The results that these mathematicians obtained were often controversial because they violated many common sense notions about equality and dimension." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"Category theory has developed classically, beginning with definitions and axioms and proceeding to a long list of theorems. Category theory is not topology (and so will not be described here), but it can be used to understand some of the relationships that exist among classes of topological spaces. It can be used to bring unity to diversity. [...] the theory of categories is not complete, it may not be completable, but it is a step forward in understanding foundational questions in mathematics." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"Contemporary mathematics is often extremely abstract, and the important questions with which mathematicians concern themselves can sometimes be difficult to describe to the interested nonspecialist." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"Every mathematical model of a scientific or engineering phenomenon must be sophisticated enough to reflect those characteristics that are important to the researcher, and the resulting equations must also be simple enough to solve." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"Excessive specialization could mean that discoveries in one discipline would have no value outside the discipline - or worse, discoveries in one branch of mathematics would remain within that branch because specialists in other branches of mathematics simply could not understand the discoveries and how they related to their own work." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"Mathematical knowledge agglomerates. When progress is made, mathematical discoveries are published for all to see, and the new is added to the old. In mathematics, new knowledge does not replace old knowledge, it grows alongside it. This is different from the way science often progresses. In science, new discoveries often replace old ones." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"Much of mathematics can be understood as “model building.” In some cases, this is obvious. Applied mathematicians, for example, routinely describe their work as mathematical modeling. […] They make certain assumptions about the system in which they are interested, and then they investigate the logical consequences of their assumptions. They are successful when their models are simple enough to solve but sophisticated enough to capture the essential characteristics of the systems in which they are interested. Some so-called pure mathematicians also build models. They examine many different mathematical systems and attempt to identify those properties that are important to their research. They isolate those properties by specifying them in a set of axioms, and then they investigate the logical consequences of the axioms." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"Of all mathematical disciplines, algebra has changed the most. While earlier generations of geometers would recognize - if not immediately understand - much of modern geometry as an extension of the subject that they had studied, it is doubtful that earlier generations of algebraists would recognize most of modern algebra as in any way related to the subject to which they devoted their time." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"The details of the shapes of the neighborhoods are not important. If the two sets of neighborhoods satisfy the equivalence criterion, then any set that is open, closed, and so on with respect to one set of neighborhoods will be open, closed, and so on with respect to the other set of neighborhoods." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"The idea behind an inductive definition is to define a set of terms sequentially, usually beginning with a trivial case, and proceeding up the chain of terms. Each link in the chain is defined in terms of the preceding link." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"[…] topologies are determined by the way the neighborhoods are defined. Neighborhoods, not individual points, are what matter. They determine the topological structure of the parent set. In fact, in topology, the word point conveys very little information at all." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"What distinguishes topological transformations from geometric ones is that topological transformations are more 'primitive'. They retain only the most basic properties of the sets of points on which they act." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"While mathematicians now recognize that there is some freedom in the choice of the axioms one uses, not any set of statements can serve as a set of axioms. In particular, every set of axioms must be logically consistent, which is another way of saying that it should not be possible to prove a particular statement simultaneously true and false using the given set of axioms. Also, axioms should always be logically independent - that is, no axiom should be a logical consequence of the others. A statement that is a logical consequence of some of the axioms is a theorem, not an axiom." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)