The empty set comes quite naturally from two principles:
(1) The ability to state what elements sets have in common.
(1a) The elements that sets have in common must always be equal to a set.
(2) That two sets might be disjoint. — fdrake
Maybe - maybe - we can sharpen this with an example. What is a warehouse? If you hold that a warehouse is a place where goods are stored, then there can be no such thing as an empty warehouse, and if you say there is, then I reject the entire idea of warehousing. — tim wood
but you each end up saying that math itself is flawed therefore there's no empty set. There must be a name for such an argument. You want to argue a very narrow technical point and your only argument is to blow up the entire enterprise. — fishfry
Math precedes foundations. Not the other way 'round. — fishfry
I came to this forum about 6 months ago because I wanted to present some ideas that in my opinion are original but are not completely formalized (until now, maybe they will be one day). I discovered that nobody understood what I meant. So probably that's the wrong approach. I am not interested in discussing ZFC. I don't know ZFC and I am not really interested in studying it.
I think I'll have to complete my work in a more formal way and present it to a math forum.
Sorry for bringing trouble. You can continue without me. — Mephist
I think I'll have to complete my work in a more formal way and present it to a math forum. — Mephist
Many people who come to math through foundations believe math is about the foundations. It's the other way 'round. Mathematics comes first and foundations are just our halting and historically contingent attempts to formalize accepted mathematical practice. — fishfry
I am just pulling these out to highlight them as excellent observations. After all, Peirce--the founder of pragmatism--was decidedly non-foundationalist in his own philosophical system. SDG/SIA captures certain aspects of true continuity that modern analysis using the standard real numbers does not.The example of synthetic differential geometry was given to show that the point of alternative foundations is to shed light on problems, not to brag about which foundation is more fundamental. — fishfry
I am just pulling these out to highlight them as excellent observations. After all, Peirce--the founder of pragmatism--was decidedly non-foundationalist in his own philosophical system. SDG/SIA captures certain aspects of true continuity that modern analysis using the standard real numbers does not. — aletheist
Math precedes foundations. Not the other way 'round.
— fishfry
How could you conceive of this? — Metaphysician Undercover
I say "there's no empty set therefore math is flawed". — Metaphysician Undercover
As applied to mathematics, yes. Charles Peirce adopted his father Benjamin's definition of it as the science that draws necessary conclusions about hypothetical states of things. There are no restrictions on conceiving such hypothetical states, other than that they be suited to the purposes for which we wish to analyze them. Applied mathematics seeks to formulate hypothetical states that resemble reality in certain significant ways, but pure mathematics does not have that particular limitation.We adopt the axioms that give us a good theory. Is that Peircean pragmatism? — fishfry
As applied to mathematics, yes. Charles Peirce adopted his father Benjamin's definition of it as the science that draws necessary conclusions about hypothetical states of things. — aletheist
There are no restrictions on conceiving such hypothetical states, other than that they be suited to the purposes for which we wish to analyze them. Applied mathematics seeks to formulate hypothetical states that resemble reality in certain significant ways, but pure mathematics does not have that particular limitation. — aletheist
It is consistent with Peircean pragmatism in the sense that one's purpose dictates how one formulates the hypothetical state of things to be explicated--in this case, the particular set of axioms to adopt.Is that an example of Peircean pragmatism or is that an expression of something else? — fishfry
I can try. As I noted several times a few weeks ago, MU employs a rigid metaphysical terminology and seeks to impose it on everyone else. "Existence" has only one meaning, and that is its ontological definition; so asserting mathematical existence is, according to MU, asserting some kind of ontological existence--no matter how many times and in how many ways we explain that this is not what anyone actually means by mathematical existence. More broadly, my impression is that MU--as the name suggests--is a philosophical foundationalist who begins with metaphysics, deriving everything else from that. Accordingly, MU has been quite dismissive of pragmatism on various occasions.... can you help me to understand Metaphysician Undercover's point of view? — fishfry
Metaphysics consists in the results of the absolute acceptance of logical principles not merely as regulatively valid, but as truths of being ... Just as the logical verb with its signification reappears in metaphysics as a quality, an ens having a nature as its mode of being, and as a logical individual subject reappears in metaphysics as a thing, an ens having existence as its mode of being, so the logical reason, or premiss, reappears in metaphysics as a reason, an ens having a reality, consisting in a ruling both of the outward and of the inward world, as its mode of being. The being of the quality lies wholly in itself, the being of the thing lies in opposition to other things, the being of the reason lies in its bringing qualities and things together. — Peirce, c. 1896
Knowledge of the history of math.
But it's the same in any discipline. There's science, and then there's the philosophy of science. One can and does do science without regard for its philosophy. That's true in every field. X precedes the philosophy of X. — fishfry
The task of the philosopher is to explain how it comes to be that math and science are abstract yet useful. — fishfry
He rejects the idea of taking math on its own terms; insists that it must refer to something outside of itself. Nobody believes that, not about pure math. Why does Metaphysician Undercover believe that? What is the basis for his ideas? — fishfry
I happen to be a practicing structural engineer. What might not be obvious to you and others is that a structure is always designed first, followed by its foundations. We have to establish what needs to be supported before we can go about determining how it will be supported. Often there are multiple options--footings on native soil, footings on special fill materials, footings on improved ground, driven steel piles, driven precast concrete piles, auger-cast grout piles, drilled shafts filled with cast-in-place concrete, etc. Typically what we end up selecting is either the least expensive solution or the one that in our judgment properly balances cost with risk. In other words, the "best" foundation can only be determined relative to a specific purpose. Why would philosophy be any different?It should be obvious to you, a structure is built on foundations, not vise versa. — Metaphysician Undercover
A philosophy of mathematics identifies the foundations that have already been established by the practice of mathematics. Most people who use mathematics, including most professional mathematicians, have little or no knowledge of its foundations--and little or no need to have such knowledge. Likewise, most people in general have little or no knowledge of philosophy in general--and little or no need to have such knowledge.The philosophy of math might study the foundations, but it does not produce the foundations. — Metaphysician Undercover
Some philosophers define truth as mere usefulness. I am not one of them, and neither was Peirce, although many of them call themselves "pragmatists"--one reason why he eventually started calling himself a pragmaticist instead.But you misunderstand philosophy, which seeks to distinguish truth from mere usefulness. — Metaphysician Undercover
In other words, philosophy is useful in relation to the goal or end of knowing for the sake of knowing. Well, so is pure mathematics in relation to the goal or end of knowing what follows necessarily from certain axioms, purely for the sake of knowing it."Usefulness" is judged in relation to a goal or end, and that goal or end may be contrary to truth ... The philosopher seeks to know, without regard for the usefulness of that knowledge, knowing for the sake of knowing. — Metaphysician Undercover
Typically what we end up selecting is either the least expensive solution or the one that in our judgment properly balances cost with risk. — aletheist
In other words, the "best" foundation can only be determined relative to a specific purpose. Why would philosophy be any different? — aletheist
A philosophy of mathematics identifies the foundations that have already been established by the practice of mathematics. Most people who use mathematics, including most professional mathematicians, have little or no knowledge of its foundations--and little or no need to have such knowledge. Likewise, most people in general have little or no knowledge of philosophy in general--and little or no need to have such knowledge. — aletheist
In other words, philosophy is useful in relation to the goal or end of knowing for the sake of knowing. Well, so is pure mathematics in relation to the goal or end of knowing what follows necessarily from certain axioms, purely for the sake of knowing it. — aletheist
Thanks for confirming that you have no idea how building design and construction actually work.What we really end up with "typically", is an extremely expensive building designed to protect the engineer from lawsuits. — Metaphysician Undercover
History demonstrates otherwise, as @fishfry has pointed out.The practise of mathematics itself, requires that a foundation already be laid in order for that practise to occur. — Metaphysician Undercover
The axioms of pure mathematics are neither true nor false, and claiming otherwise is a category mistake. If a certain theorem follows necessarily from a particular system of axioms, then it is true within that system.If the axioms involved are false, then what follows from them cannot be sound conclusions. — Metaphysician Undercover
History demonstrates otherwise, as fishfry has pointed out. — aletheist
The axioms of pure mathematics are neither true nor false, and claiming otherwise is a category mistake. — aletheist
LOL! Right back at you.One assertion after the other, without explanation or justification, together with an ignorance of logic; you just reminded me of how difficult it is to hold a discussion with you. — Metaphysician Undercover
The practise of mathematics itself, requires that a foundation already be laid in order for that practise to occur — Metaphysician Undercover
Assuming that by "foundations" you mean the acceptable accumulated knowledge and practices up to any particular moment of mathematical history. Weierstrass and Cauchy laid the critical foundations for my interests, above and beyond what came before. — jgill
I can try. As I noted several times a few weeks ago, MU employs a rigid metaphysical terminology and seeks to impose it on everyone else — aletheist
Notice that no conceptual structure is completely new, in an absolute sense, it's always built with some already existing principles. — Metaphysician Undercover
So MU is an anti-Peircean. That actually helps me understand Peirce. Sounds like I'm a Peircean and never knew it. — fishfry
How would you know? You have not demonstrated any familiarity with his voluminous writings.Furthermore, I actually agree with a lot that Peirce has said. — Metaphysician Undercover
Nonsense. As usual, bare assertion with no basis in fact.Aletheist resorts to contradiction in an effort to support Peirce's illusory solutions ... — Metaphysician Undercover
I refuse to acknowledge this because it is blatantly false.... because Peirce draws on dialectical materialism, or dialetheist principles which decline ruling out contradiction. Aletheist refuses to acknowledge this. — Metaphysician Undercover
That's not a logical conclusion.
Furthermore, I actually agree with a lot that Peirce has said. in particular his very coherent method of laying out particular epistemological problems. What I disagree with is the metaphysics he proposes to resolve those problems. He provides no real solution, only the illusion of a solution. Aletheist resorts to contradiction in an effort to support Peirce's illusory solutions, because Peirce draws on dialectical materialism, or dialetheist principles which decline ruling out contradiction. Aletheist refuses to acknowledge this. — Metaphysician Undercover
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