Mathematics may be defined as construction game leading to a big set of self-coherent intellectual entities : they do not have any existance outside of our head (no herd of "Twos" in the woods). This pure intellectual construction is mainly made by strange humans (called mathematicians) with no care of applications (except some exceptions). From this intellectual construction, other people (unbelievable but true) pick some maths entities and a priori decide to match them with some real world observations. These strange kind of people are called physicians, chemists, ... and applied maths engineers. We show in the next figure the conceptual links between several maths-based human activities that lead together to what is generaly called a 'mathematical model' :
NB : Human being is the key element of main items in this scheme : - Observing a part of the Real World through a finite number of sensors with finite resolution and range is a human activity : what to observe, using which sensors, why, ... are questions that find answers in a priori knowledge and belief of humans. For 'the same Real World', the choice of different experiments and sensors may lead to different observations (and then to different mathematics/observations matching). - Building mathematics as a self coherent set of entities (what we could call 'pure maths'), discussing about what "self coherent" means, about what "demonstrated", or "exist" means, ... is a human intellectual activivity : ex : is it possible to create ex nihilo an entirely coherent system without a priori ? ... is a question that led to define the "axiom" notion (cf. the axiom of choice) that is the mathematical word for a priori knowledge and belief. - Choosing to fit observations into pure maths entities, and then use inheritance of their properties and their ability to combine in order to build new entities, is a human activity using a priori knowledge and belief : ex : 'space' and 'time' are not fit into the same mathematical entities in Newton or in Einstein Physics ... does that mean that 'space' and 'time' properties have changed between 1665 and 1920 ? One can notice that experiments and observations techniques made big progress between those two dates ! and getting new observations of variations gave new 'ideas' of matching ... and led to new mathematical models.
Once every human activity has been done, then, we get MATHEMATICAL MODELS that are usually described in Universities as entirely self-coherent disciplines with no human intervention (and that is true : human intervention was to create them. Once model created, then inheritance allows to talk about observed entities with a vocabulary derived from pure maths, using formal combination operations ...). But it is important not to forget that : - observations of variations ARE NOT the real world
- models ARE NOT the real world AT ALL
- models ARE NOT pure maths
Some facts that show their difference :
- observations give a representation of the real world, compatible with our senses (and mainly vision : we are can handle a 1, 2, or 3 D representation of an observation, not more !) in a finite range of precision, bandwith, ..., outside of this range, no one "knows" what's going on. - mathematical models INTRINSICALLY produce ERRORS (of prediction/estimation ...) : once error is lower than a given value, one can say that the MODEL IS "TRUE" (a very different definition of truth than in the pure maths world !). NB : EVEN if the error seems to be null ... one should NEVER consider that the model is "perfect" because : - measurements have a finite precision (so what a "null error" means ?) - in practice, there always stays "small" unexplained variations called "noise" - comparison between prediction of the model and observations was made ONLY in a finite number of cases - experiment modifies the part of the real world that one tries to...