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B What are Constructs in Mathematics?

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  1. Nov 30, 2017 #1
    Is Set Theory a Construct? If yes, then what else are Constructs in Mathematics? Some website defines the word Construct as "something formulated or built systematically". I want to understand the meaning of the word Construct. Is the word Construct in mathematics same as the word Construct in Computer Programming Languages?
     
    Last edited: Nov 30, 2017
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  3. Nov 30, 2017 #2

    QuantumQuest

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    Gold Member

    Construct is something made of a number of simpler elements and this holds true for both Mathematics and software that you ask. But the way you state it, it is unclear what exactly you ask for.
     
    Last edited: Nov 30, 2017
  4. Nov 30, 2017 #3

    jedishrfu

    Staff: Mentor

    Here's an article that might answer your question:

    http://www.sciencedirect.com/science/article/pii/S0883035597884443


    Mostly this comes up in an educational setting.
     
  5. Nov 30, 2017 #4
    Thank you for the replies.

    If I look at a computer programming language, like C, it has 'if' construct, it has 'conditional' construct, it has 'class' construct.
    My question is: if I talk about constructs in Mathematics then am I correct to say that Set Theory is a mathematical construct; Functions in Set Theory are mathematical construct? If yes, then what are other mathematical constructs in mathematics. Is mathematics itself a construct?

    The definition of construct is: something constructed by the mind as
    a: a theoretical entity
    b: a working hypothesis or concept
    c: a product of ideology, history or social circumstance

    Thanks
     
  6. Nov 30, 2017 #5

    jedishrfu

    Staff: Mentor

    Yes, you could say that Mathematics has been invented by humanity ie constructed by us as an aid to understand the world. We started with numbers and then the number line to number plane to n-dim...

    We apply these constructs to the physical and found we can make predictions. We apply them to themselves and create new mathematics.

    One great example is origami paper folding where its been shown that origami can solve the famous doubling the cube and squaring the circle problems of antiquity.
     
  7. Nov 30, 2017 #6

    fresh_42

    Staff: Mentor

    What you call a construct, could be viewed as a category. A category is basically a set of objects (sets, groups, vector spaces, fields, etc. etc.) and the mappings between those objects, which preserve the structure, e.g. linear mappings in case of vector spaces, any functions in case of sets. These are close to what a class is in C. The constructor would be an instance of a category (a certain set) and the functions are the methods of the class. The attributes of a class are then the attributes of the category, e.g. the group axioms in case of the group category.

    Whether mathematics as a whole is a construct, is a purely philosophical question. E.g. a perfect circle doesn't exist in reality, but we have no problems to deal with it mathematically. So you may call it a construct, but this is more a question of taste or the philosophical school you follow.
     
  8. Dec 1, 2017 #7

    QuantumQuest

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    Gold Member

    I see two things here that must be further clarified, in order to not cross the boundary between science / technology on the one side and philosophy on the other.
    The first is if we talk about physical or conceptual constructs. The second is how do you utilize the word construct, meaning where exactly do you put the boundaries of something being a construct or not. If we don't put any kind of boundary then we'll end up to the philosophical realm.

    In Mathematics there is obviously no physical construct so we talk about axioms, theorems and all the ingredients that make up the world of math. You can say that all these are conceptual constructs but in my opinion the term concepts would be more appropriate for some things and constructs for other. Now, anyone can argue that even concepts are in many cases constructs themselves but following this perpetual path you'll eventually cross the boundary of mathematics and get into philosophy. For mathematics themselves as a whole, I'll just quote what @fresh_42 said as this is my opinion too

    Now, talking about a programming language like C that you mention about, it has conditional constructs like "if" in conceptual form i.e. regarding the design of the language but it also has these conditional constructs implemented using grammar and syntactic rules inside a compiler as well - C has no class concept; you probably refer to C++ for this but the idea about constructs is the same. Going further, if you develop some software product then this as a whole is a construct that is utilizing other constructs, for instance smaller programs, that themselves utilize some programming language's constructs and so on and this holds true both for the conceptual level regarding the design of the product as the implementation i.e the product you install on a computing machine as well.

    So, I think that it becomes obvious that while it is not wrong to use the term "construct", it is a very general and abstract term that needs further clarification at various levels, depths and widths and does not give sufficient clarity in many cases.
     
    Last edited: Dec 1, 2017
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