I have a question about the nature of generalization and abstraction. Human reasoning is commonly split up into two categories: deductive and inductive reasoning. Are all instances of generalization examples of inductive reasoning? If so, does this mean that if you have a deep enough understanding of inductive reasoning, you broadly create “better” abstractions?
For example, generalizing the integers to the rationals satisfies a couple of things: the theoretical need to remove previous restrictions on the operations of subtraction and division, and AFAIK the practical need of representing measurable quantities. This generalization doesn’t seem to fit into the examples given here http://en.wikipedia.org/wiki/Inductive_reasoning at first glance, and I was hoping someone could give me some nuggets of insight about this. Or, can someone point out what the evidence is that leads to this inductive conclusion/generalization?
A couple of starting points to improve your investigation.
First, rigorous inductive reasoning, i.e. bayesian probability, includes as a special case deductive reasoning, at least in the case where deductive reasoning is conflated with “classical logic”. AFAIK there have been only sparse and timid research into widening probability to fit other kinds of logic.
Second, the example you use to illustrate generalization is a case of what in logic is known as elementary embedding, and it’s a pure application of deductive reasoning. Although the process that led the first mathematician to the invention of rational numbers might have very well been one of inductive reasoning.
Third, do not be tempted to think that abstraction is generally linear. It may very well be the case that the model C is generalized by model B, which is generalized by model A which is found to be a special example of C. This kind of “strange loops” (I credit Hofstadter for the invention of this term) happens all the time in category theory. The very same four different foundations for mathematics (set theory, type theory, category theory, univalence theory) are all instances of one another (see for example the paper “From Sets to Types to Categories to Sets” by Awodey, which however was compiled before univalence was formalized).
Do you have any recommendations of what to study to understand category theory and more about the foundations of math? (Logic, type theory, computability & logic, model theory seem like contenders here)
You’re welcome! Foundation(s) of math is a huge and fascinating topic by itself, but if you’re more interested in the intricacy of abstraction hierarchies you should look no further than category theory, the Yoneda lemma, up up to doctrine theory. I love as a good introduction Category Theory by Awodey. As for the foundation of math, very good for dipping your toes are the first chapters of Marker’s introduction to model theory and the recently reprinted Set theory by Kunen (set theory models are a vast subject by themselves...)
I have a question about the nature of generalization and abstraction. Human reasoning is commonly split up into two categories: deductive and inductive reasoning. Are all instances of generalization examples of inductive reasoning? If so, does this mean that if you have a deep enough understanding of inductive reasoning, you broadly create “better” abstractions?
For example, generalizing the integers to the rationals satisfies a couple of things: the theoretical need to remove previous restrictions on the operations of subtraction and division, and AFAIK the practical need of representing measurable quantities. This generalization doesn’t seem to fit into the examples given here http://en.wikipedia.org/wiki/Inductive_reasoning at first glance, and I was hoping someone could give me some nuggets of insight about this. Or, can someone point out what the evidence is that leads to this inductive conclusion/generalization?
A couple of starting points to improve your investigation.
First, rigorous inductive reasoning, i.e. bayesian probability, includes as a special case deductive reasoning, at least in the case where deductive reasoning is conflated with “classical logic”. AFAIK there have been only sparse and timid research into widening probability to fit other kinds of logic.
Second, the example you use to illustrate generalization is a case of what in logic is known as elementary embedding, and it’s a pure application of deductive reasoning. Although the process that led the first mathematician to the invention of rational numbers might have very well been one of inductive reasoning.
Third, do not be tempted to think that abstraction is generally linear. It may very well be the case that the model C is generalized by model B, which is generalized by model A which is found to be a special example of C. This kind of “strange loops” (I credit Hofstadter for the invention of this term) happens all the time in category theory.
The very same four different foundations for mathematics (set theory, type theory, category theory, univalence theory) are all instances of one another (see for example the paper “From Sets to Types to Categories to Sets” by Awodey, which however was compiled before univalence was formalized).
Thanks for your enticing comment!
I understand your first point, but my math knowledge is not up to par to really understand point #2, and point #3 just makes me want to learn category theory. BTW, I also posted this question on the philosophy stackexchange: http://philosophy.stackexchange.com/questions/14689/how-does-abstraction-generalization-in-mathematics-fit-into-inductive-reasoning.
Do you have any recommendations of what to study to understand category theory and more about the foundations of math? (Logic, type theory, computability & logic, model theory seem like contenders here)
You’re welcome!
Foundation(s) of math is a huge and fascinating topic by itself, but if you’re more interested in the intricacy of abstraction hierarchies you should look no further than category theory, the Yoneda lemma, up up to doctrine theory. I love as a good introduction Category Theory by Awodey. As for the foundation of math, very good for dipping your toes are the first chapters of Marker’s introduction to model theory and the recently reprinted Set theory by Kunen (set theory models are a vast subject by themselves...)