Philosophy of mathematics
This is intended to become the script for a course on the Philosophy of Mathematics. It may then serve as basis for books and other materials as well as discussions and research. Also see the discussion page of this module.
About Philosophy of Mathematics
This is an introductory meta chapter on the topic and the course of Philosophy of Mathematics (PhilM). It might be helpful for deciding on taking this course as well as for understanding its approach.
- The Topic PhilM: What is Mathematics? How is Philosophy seen here? Who is PhilM for? What is PhilM about?
- The Course PhilM: didactics, boundaries, ...
- The Topic PhilM: What are the structuring principles? What does the structure look like?
- The Course PhilM: didactics, approach, ...
- Defining Mathematics - The subject usually called Mathematics.
- Other Basics used within the course - misc. basics can be added here ...
Understanding the historical development of PhilM ...
Philosophy of Supplying Mathematics
A Mathematician is thinking about the degrees of freedom he or she has while doing his or her job, finally starting to question the foundations of a mathematician's job.
Questioning the Method
Judging mathematical work beyond right and wrong: How to do the Mathematician's every day job? How to exploit the degrees of freedom a Mathematician has in doing a Proof, judging the importance of Sentences and overthrowing Theories?
- Questioning the Instance of Proof - The Exam Situation. Why do I do my Proof the way I do? What is a good or bad proof? Is a proof an explanation? ...
- Questioning the Sentence - Normal Science. What Sentence to prove? What is an important Sentence? ...
- Questioning the Theory - Revolutionary Science. How do Theories interrelate? What is an important Theory? ...
Questioning the Foundations
How well-founded is Mathematics? Is there a better Mathematics than the one we employ today?
- Questioning the Concept of Proof - Metamathematics. Is there a mathematical definition of a mathematical proof?
- Questioning the Language - Metamathematics. What can and what can't be expressed with the usual mathematical Language? Are there more expressive alternatives? Does increased expressiveness come for free? ...
Philosophy of Applying Mathematics
Or "Philosophy of Logic"
Questions concerning justification and use:
- How can a Theorem/Theory be of any use in general?
All theorems are either (a) an assumption agreed upon by a posteriori knowledge (something we intuitively observe), or (b) a priori knowledge. (something we assume is true within a rational language by strict rules of inference).
In other words, theorems are either facts we can generally agree on about the way the universe works around us (If you put one apple in a box with one other apple, the box will have 2 apples), or they are concepts which we base a formal language on (1+1=2).
Deductive reasoning (A priori): By collecting a vast amount of theorems, we can use these theorems together and derive / dedeuce other assumptions about the world around us. Ie: We can take different things that we all assume, and use them together to assume more things. Eg: "If a+b=c, and c-b=a, then c-a=b"
Inductive reasoning (A posteriori): By collecting a vast amount of theorems, we can connect some of our assumptions and approximate highly probable theories about how they function together. Eg: "Out of every of the thousands of DNA strands we've examined on Earth, every one of them contains the base-pairs of Adenine, Guanine, Thymine, Cytosine. So it is fair to assume that with possible rare exceptions- all DNA strands on Earth have the base pairs of Adenine, Guanine, Thymine, Cytosine"
- How beneficiary is the Theorem/Theory, esp. to science and engineering?
- Where and how may one apply this Theorem/Theory?