What Is Maths Model
WHAT A MATHEMATICAL MODEL IS NOT
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
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