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Foundations of Mathematics

  1. Jul 2, 2009 #1
    I`ve started to read on FOM, and unfortunately I found the first paragraph cumbersome to get.

    Here is the text I started with: http://www.math.psu.edu/simpson/hierarchy.html [Broken]

    The first paragraph was:
    "1. All human knowledge is conceptual and forms an integrated whole. All human knowledge is contextual and hierarchical."

    The three radical words are: conceptual, contextual, and hierarchical.

    I understand what "hierarchical" means. I`m gonna get back to "contextual" later.

    Now what I`d like to understand is the word "conceptual", so can I get help here?

    I did some researching about "conceptual", and one good website was: http://www.education.com/reference/article/distinction-conceptual-procedural-math/
    But no single website gave a precise definition of what "conceptual" means.. which makes me scratch my head really hard trying to understand based on what did the author of the FOM text use the word "All" at the beginning of the text, and what was he thinking of when he used the word "conceptual".
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Jul 2, 2009 #2
    That essay is itself part of the philosophy of mathematics, not the foundation of mathematics. It is the author's explanation of the role and definition of "foundations of mathematics", and thus as I said the essay itself really belongs to philosophy of mathematics.

    If you are interested in the foundations of mathematics, then you would be pointed towards mathematical logic and axiomatic set theory and/or category theory. These subjects are extremely rigorous and would never contain such a vague statement as:

    "1. All human knowledge is conceptual and forms an integrated whole. All human knowledge is contextual and hierarchical."

    except perhaps in the preface, or in the end of chapter notes /motivational material. Such a statement lies clearly in the domain of th philosophy of mathematics, which is a sub-branch of philosophy, and so bears little resemblance to mathematics itself.

    To get an idea about what FOM is all about it might be better to glance through that same author's publication list (although the library is a better place to start):


    The statement about "all human knowledge..." is an attempt by a mathematician to go outside of his specialty and do philosophy, and so the most polite thing I can say is that I did not find any of the statements in that essay to be worthwhile.
  4. Jul 2, 2009 #3


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    "Conceptual": based on concepts. What is being said here is simply that basis of all knowledge is the "concept" or "idea".

    He was saying that all knowledge is based on "ideas". Although without reading the whole thing I can't say for sure, I suspect he was thinking "as opposed to sense impressions".
    Last edited by a moderator: May 4, 2017
  5. Jul 2, 2009 #4
    So, it seems like Philosophy of Mathematics is more fundamental than Foundation of Mathematics..
    Well, the reason I started with FOM is that I wanted to start from the very core of mathematics, and I thought FOM was the most fundamental thing to start with, but since you are saying that "that essay is itself part of Philosopgy.." I think I should start with Philosophy?
  6. Jul 2, 2009 #5
    I understand what the word "concept" itself mean; but, I thought it might be used not literally, and the first question that motivated me to come and ask here is: "how do we know that "All" knowledge is conceptual?"

    What I`ve just learned from Mr.Civilized is that I don`t have to take the text very literally; and from you, sir, I understand he meant the literal meaning of "concept".
  7. Jul 2, 2009 #6


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    No, not at all. It is used to describe mathematics, but the math does not rely on it.

    Now it could be argued that philosophy (or at least some branch of philosophy) is more fundamental than mathematics, but philosophy of mathematics is certainly not.
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