- Mathematics is the study of discrete or continuous quantity;
- Mathematics is not the study of all relations.
- Semiotic is the general science of relations (signs).
- Physicist and America's greatest philosopher Charles Sanders Peirce said (CP 5.448) that "this universe is perfused with signs, if it is not composed exclusively of signs."
- Boethius's famous definition of mathematics is (De Trinitate q. 5):
"Mathematics does not deal with motion [change] and is not abstract [for this sense of abstract, see Armand Mauer's introduction to his translation of Thomas Aquinas's Division and Methods of the Sciences], for it investigates forms of bodies apart from matter, and therefore apart from movement, which forms being connected with matter cannot really be separated from bodies."
- Thomas Aquinas, in Division and Methods of the Sciences q. 5 a. 3, argues that mathematics treats "without motion [change] and matter, of what exists in matter."
- See also this question's 5th objection, regarding how
"mathematics treats without motion," and his reply to it, where he says
"it does not belong to the mathematician to treat of motion, although
mathematical principles can be applied to motion." He also shows
mathematical physics is an "intermediate science" (scientia media), since it's materially physical and formally mathematical.
- Cf. Peirce's "The Nature of Mathematics," ch. 10 (p. 135) of The Philosophy of Peirce.
- quantity does not comprise the substances of material things;
- Aristotle writes (Metaphysics 1017b16-22):
"substance means those parts which, being present in such things, limit them and designate them as individuals and as a result of whose destruction the whole is destroyed; for example, body is destroyed when surface is, as some say, and surface when line is. And in general it seems to some that number is of this nature; for [according to them] if it is destroyed, nothing will exist, and it limits all things."Thomas Aquinas commentates (Sententia Metaphysicae, lib. 5 l. 10 n. 3 [900-1]):
"He gives a third meaning of substance, which is the one used by the Platonists and Pythagoreans. He says that all those parts of the foregoing substances which constitute their limits and designate them as individuals, according to the opinion of these thinkers, and by whose destruction the whole is destroyed, are also termed substances. For example, body is destroyed when surface is, as some say, and surface when line is. It is also clear that surface is the limit of body and line the limit of surface. And according to the opinion of the philosophers just mentioned the line is a part of surface and surface a part of body. For they held that bodies are composed of surfaces, surfaces of lines, and lines of points; and thus it would follow that the point is the substance of the line, the line the substance of surface, and so on for the rest. And according to this position number seems to constitute the entire substance of all things, because when number is destroyed nothing remains in the world; for what is not one is nothing. And similarly things which are not many are non-existent. And number is also found to limit all things, because all things are measured by number.
"But this sense of substance is not a true one. For that which is found to be common to all things and is something without which they cannot exist does not necessarily constitute their substance, but it can be some property flowing from the substance or from a principle of the substance. These philosophers also fell into error especially regarding unity and number because they failed to distinguish between the unity which is interchangeable with being and that which is the principle of number."
- rather, quantity presupposes a substance: "of all the accidents [properties of a substance] quantity is closest to substance" (Sententia Metaphysicae, lib. 5 l. 15 n. 7 ).
- ∴ everything is not mathematics.