# The demarcation problem of mathematics

The demarcation problem in the context of philosophy is usually used to mean the demarcation problem of science, the problem of separating science from non-science. However, what about the demarcation problem of mathematics, that is, the problem of separating mathematics from non-mathematics? Has any philosopher talked about that? And if so, can I see some references for their discussion? I am particularly interested in the question of how to decide whether some statement is mathematical or not, and also how to decide whether some entity is mathematical or not. For instance, most people agree that the number 3 is a mathematical entity, and that a wooden desk is not a mathematical entity, but how exactly does one demarcate the boundaries of math?

• The "demarcation problem" is not about "separating science from non-science" but about Science vs Pseudo-science: we have no problem in understanding the difference between science form one side and music, football, religion from the other side. Jan 31 at 7:52
• Thus, a similar issue regarding mathematics may involve e.g. numerology; in general, what counts for a statement to be mathematics or not is the opinion of mathematical community. Jan 31 at 7:54
• See pseudomathemtics:"a mathematics-like activity that does not adhere to the framework of rigor of formal mathematical practice." The term was coined by de Morgan. The demarcation problem for mathematics is considered uninteresting for its triviality, the rigor standard and its application in practice are more or less consensual among experts. Jan 31 at 8:12
• A simple approach you can use for your example: a wooden desk is perceived by the senses, the number 3 is a rational outcome. This is moreover the Kantian approach: mathematical notions would be a priori to experience (e.g. reason, that is, we invent the circle, there are no circles in nature, only polygons); knowledge of the world would be a posteriori from experience (i.e. use of the senses is imperative, ẃe can't know wood without the senses). Jan 31 at 8:29
• @Conifold - before the 19th century, most mathematicians were involved in "pseudo-mathematics", given the standard of rigors they were using :-) Feb 1 at 4:17

Your question is analogous to asking how exactly does one demarcate the boundaries of culture. The answer is that you cannot define the boundary exactly in a way that everyone would necessarily accept. Indeed, I would ask you to say why you consider that it should have a boundary. Clearly there are distinctive characteristics of mathematics, which you could readily list, but the practical applications of mathematics are potentially endless and vary enormously according to the type and degree of mathematical content. If I am an economist, I might be developing mathematical relationships between certain attributes of products and their prices, where mathematical symbols appear recognisably in my work. If I am a carpenter working out how to make a complicated joint, I am using a form of applied mathematics, considering angles and lengths and their geometric relationships in the abstract, but possibly without any recognisable use of mathematical symbols. If I am a chef, I might be concerned with the quantity and proportions of the ingredients in my menu, which is a mathematical consideration. It is possible to imagine a multi-dimensional continuous spectrum of activities differing by the nature and degree of their mathematical content, and any attempt to define a precise boundary between the parts of that conceptual volume that are mathematics and the parts that are not must essentially be arbitrary.

You might have more luck in attempting to define a boundary if you were to limit your definition of mathematics to include only pure mathematics, but even then I suspect there will be undecidable borderline cases where arguments could be made for a given example to be pure or applied.

• culture have it's historical semantic borders. Culture term becomes at XIX century from agriculture, and as science discipline came only at XX. Math date appearing is unknown. Culturology has it's spectre of problems, math problems are compare with logic and braine functions. Not need to have 2 "maths" in one area. Jan 31 at 6:33

There is a philosophical task connecting pure mathematics to the real world. There is a jump from the abstract real line, which is one thing, and the range of allowed speeds of a particle, which is another thing entirely. Where does the maths end and science begin?

This was a controversial topic in the 20th century, at the time of David Hilbert and his school of mathematical formalism. This is around the time of Russel's Paradox, and the broader program to formalise mathematics with axioms everyone agrees on.

Other keywords include Platonism, positivism, constructivism, intuitionism. Important names include Poincaré, Cantor, Brouwer, Godel, Russel.

For a rollicking introduction to the topic I recommend reading Logicomix:

Math is full abstract model of logic: in real world there is no 2 equal objects, or 2 synchronous moments. But human braine can exam only similar or associations. So, don't use braine for exam and that ll be math borders. It can be narrative, or broken prediction text like mental disorders, or wandering, or poetry, or axioms, or principles.

• "Math is full abstract model of logic." What does it mean? Jan 31 at 12:53
• "In real world there is no 2 equal object" False; electrons are all equal. Jan 31 at 12:53
• "It can be narrative, or broken prediction text like mental disorders" What does it mean? Jan 31 at 12:54
• i mean your-mine second notice. Okey, if you address to microcosm physics, you should to explain what is an electron is it material or wave? are you have every electron in same moment and same space coordinates? If particles haven't different temperature, if you stop their moving and you got superfluid liquid where all is similar, and you can't separate one particle from the drop or else( my fantasy). It is similar homogeneous substance if you have similar energy. So you got something strange, but not 2 equal but different objects. So you need space, time and energy level to detect difference Jan 31 at 13:27
• You need to do abstraction "cut" of some similar thing attributes, and don't pay attention at differences of things. Braine work this way, it "cuts" that he need, from full object. When you say electrons are equal - that mean some attributes of electrons are equal. About narrative. Narrative it is not story, narrative it is details that are predict to the story. You have some details: cube, pin, black and ferret. This is narrative. You can create the story from this details by placing them in some cause or some accents. If you change order of narrative's details - you got another story. Jan 31 at 13:47