Philosophy-6 Flashcards
Mathematical Sciences VS Experimental Sciences
Classical separation of Sciences:
●Mathematical Sciences is a priori or deductive.
●Experimental Sciences is a posteriori or inductive.
Mathematical Sciences (arithmetic, algebra, geometry, trigonometry, and calculus)
The study of quantities and relations among them, by using numbers and conventional symbols. It is abstract:●continuous as in shapes, surfaces and volumes.
●discontinuous as in numbers.
The practical development of Mathematics
Developed as an expression of the practical human need to count things in the surroundings and to measure time (primitive methods: using fingers, pebbles, carving on walls..).—> into mathematical concepts, symbols, shapes, postulates, equations…—>the stage of computer use.
This development led to the birth of 2 claims/theories:
●The Empiricists: claim that mathematical notions have been born out of experience.
●The Rationalists: claim that Mathematics is a field independent from reality and us idealistic in nature.
Problematic of Mathematical Sciences
What is the origin of mathematical notions? Are they idealistic and born from pure reason? Or are they born from sensory experiences?
The Empirical Perspective
●believe that mathematical notions, like any other knowledge, are derived directly from sensory experiences (tabula rasa)—> John Stuart Mill’s System of Logic follows this perspective.
●argue that Mathematics did not appear with man but rather accommodated his needs later on as an inductive science starting from the sensed parts to reach concepts and theories (evident in ancient civilizations that used maths for architecture and landscaping).
●believe that numbers have a utilitarian value (for counting), and that the origin of the notion of a number is the empirical perception of a collection of objects.
Benefits of the Empirical Perspective on Mathematics
It has been successfully applied in education by using sensory effects, due to it’s efficiency in teaching students by approaching real experiences and activities. Also by using visual aids to teach shapes, numbers, or relations..
System of Logic (John Stuart Mill)
Primitive train of logic/thought
Ex: a horizon or stuck suggested the straight line, the countour of the sun suggested the circle, the trunk of a tree suggested the cylinder..
Criticism on the Empirical Perspective
May be criticized for their concentration on the role of the senses, and underestimating the role of the mind and reasoning (priori ability ignored).
They may also be criticized for ignoring the abstraction and generalization of mathematical numbers and symbols and the ability to create them without referring to the senses (the number 0 & negative numbers & imaginary numbers & symbols…).
The Rationalist Perspective
●Defends the idealistic thesis and claims that the object of Mathematics is purely ideal, theoretical, and abstract (exists in spirit befire experience)–>it is independent of facts and doesn’t need real objects.
(Ex: PLATO says that “five” is not a thing, but merely a way of thinking about a group of things).
●Plato considered that non-physical forms (ideas) represent the most accurate reality—> they are the real world that all material strives to imitate.
●Believe that experience is only a reason to create a mathematical notion.
(Ex: Immanuel Kant says that in experience, place are limited and have dimensions, whereas mathematical place is flat, homogeneous and infinte).
René Descartes (a rationalist)
Claims that mathematical notions are innate—>it goes further into abstraction and away from sensible reality—> they follow a logical deduction.
(Nothing in reality represents zero or infinity..)
Basically, the rationalist perspective says that we don’t go to a laboratory to prove maths by experiment and observation, because it is a pure logical science not related to concrete, materialistic realities.
(On the other hand, geometry is conditional deductive science: if we accept definitions, then we accept the results that follow)
Criticism on the Rationalist Perspective
May be criticized for only considering the hypothesis as the only rational factor in the stages of scientific method. But it is also composed of ration processes and constructions that are continuous within a method.
Also might be criticized sometimes for lack of verification (through experiments) of a process because they are also the work of mind: introducing a hypothesis only a subjective suggestion, until they are metgodologically proved.
Conclusion on Mathematical Sciences
Nature manifests it’s phenomena to senses, and humans observe them. While experiment and observation direct the mind, the mind tries to suggest reasons through hypotheses. Hence, the stages of the scientific (experimental) method are 3: observation, hypothesis, and experiment.
These might overlap but they work together to solve the contradictions between science and nature.
This leads to human progress through the satisfying process to know and to build. This progress requires planning, financing, and cooperation om many levels (individual + universities and governments) to enhance humanity.
