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somethoughts - “Finitary” mathematics versus...

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Unformatted text preview: “Finitary” mathematics versus Mathematics of “infinity” Periklis A. Papakonstantinou ? University of Toronto This is an informal discussion regarding the distinction of Discrete Mathematics from other branches of Mathematics. Most Discrete Math texts begin by trying to justify what is the distinction of Discrete Mathematics from other areas like Calculus or Linear Algebra. For this they get into dis- cussions around the term “Discrete”. The author 1 of this manuscript fails to understand the notion of “concrete” as a primitive mathematical concept. To me it seems that this notion of “concrete/discrete” is a consequence rather than a primitive mathematical no- tion. What does “discrete/concrete” mean? Is the set ∅ something discrete. I think yes. Is the set {∅} discrete? How about the set {∅ , {∅}} ? The vast majority of people would say that all of the above are discrete objects. Those with some knowledge in Axiomatic Set Theory they do know that the building blocks of every mathematical system are such dis- crete objects. So what is in “Discrete Mathematics” different than in “Calculus”. I believe that the right concept here is that of finite . This is because every finite set like { 1 , 2 , 3 , 4 } involves something discrete whereas a set like (0 , 1] seems to have something quite non- discrete in its definition. It seems like an exception to this is the set of natural numbers. But this is still not true. In many common places where the infinite set of natural numbers appears in Discrete Mathematics, this set is being used because of its elements, not because of itself. Having mentioned the set (0 , 1] let me also mention another set [ 1 n , 1], for some n ∈ N . We observe that [ 1 n , 1] ( (0 , 1]. Note that the set (0 , 1] contains all real numbers that can get arbitrarily close to zero, but they never get to zero. Choose n = 10. Then, [ 1 10 , 1] = [0 . 1 , 1] ( (0 , 1]. Now choose n = 100. Then, again [0 . 01 , 1] ( (0 , 1]. Furthermore, [0 . 1 , 1] ( [0 . 01 , 1]. That is, choosing (discrete) values for n we can get as close as we want to the whole set (0 , 1] but no single n ∈ N is sufficient to get us to (0 , 1]. In a first course in Calculus students learn that the limit ( at infinity) of the sequence h 1 , 1 2 , 1 3 ,... i is zero....
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