On Apr 22, 2014, at 4:18 PM, Bruce Schuman <bruceschuman@xxxxxxx> wrote: (01)
> Wikipedia says more or less the same thing –
>
> http://en.wikipedia.org/wiki/Axiom
>
> “An axiom, or postulate, is a premise or starting point of reasoning. A
>selfevident principle or one that is accepted as true without proof as the
>basis for argument; a postulate. As classically conceived, an axiom is a
>premise so evident as to be accepted as true without controversy.”
>
> “No explicit view regarding the absolute truth of axioms is ever taken in
>the context of modern mathematics, as such a thing is considered to be an
>irrelevant and impossible contradiction in terms.”
>
> *
>
> In the context of our recent discussion here of “ambiguity” – I suppose
>my instinct is to doubt anything postulated as an axiom that does not have an
>absolutely unambiguous interpretation. (02)
What exactly do you mean by "unambiguous interpretation"? Take a real example
from mathematics: the axioms for elementary group theory. These are a set of
equations between terms written using the group product, which I will write as
an infix dot, and the inversion operator, which I will write as a prefix : (03)
a.(b.c)=(a.b).c
a.1=1.a=a
a.a=a.a=1 (04)
Now, are these unambiguous? They are in the sense of being quite clear and
exact; but not in the sense of having only one possible interpretation, since
there are infinitely many groups; which is, of course, one of the reasons why
these axioms are interesting. (05)
>
> Based on the broader definition of axiom – Richard’s axioms from Ayn Rand
>meet the criteria. In that context, what an axiom is becomes a stipulative
>definition. “An axiom is what I say it is – and if I can get people to go
>along with me, then we’ll all agree it’s an axiom” – and build our
>world on this basis – and the question of whether this is realistic or sound
>– becomes a matter of opinion. (06)
My issues with Richard's axioms are that they have no specified meaning, so it
is impossible to know how to build anything on them. The question of their
veracity or otherwise can only be raised when they have some actual meaning. (07)
> But for me – axioms with multiple alternative interpretations are
>confusing, and seem likely to lead to more confusions. (08)
Do think that group theory is confused, or confusing? (09)
> I sometimes wonder if Dedekind’s program to prove the consistency of
>mathematics (010)
Actually that was Hilbert's program, but ... (011)
> – generally considered proven impossible by Kurt Goedel – actually ran
>afoul of something inherently ambiguous in the foundations of arithmetic –
>the Peano axioms. (012)
Do you mean the secondorder or firstorder axiom set? The secondorder Peano
axioms are categorical : they have only one interpretation (up to isomorphism).
The firstorder version is of course not categorical (by Goedel) and allows
nonstandard models. All this has been known and throughly understood for close
to a century: there is no mystery here. (013)
>
> This is a heady topic – and I don’t have a clearcut direction – except
>to wonder whether it might be possible to remove any possible ambiguity in a
>structure like “A + B = B + A” by “constructing” all these elements in
>a “machine space” that is 100% explicitly known in every dimension. (014)
In "machine" implies computability, then the answer is no. (015)
>
> For me – this issue gets into the kinds of things Richard Hofstadter was
>talking about years ago in Godel Escher and Bach – and later, in Metamagical
>Themas.
>
> If the existence of a mathematical object – like “A” – is actually
>“a label for something” – how is that label constructed, and what is
>that something, and where is it located, and how is it possible to execute
>actual operations on that something? (016)
The *existence* is a label for something?? I can't make sense of this. (017)
Constructing labels is easy. Every time you type a word you create a label. (018)
> For years, I was staring at Hermann Weyl’s quote: “Nobody can say what a
>variable is”. (019)
Can you give the exact citation of this? I see it requoted all over the Web,
but I can't find the actual source. I suspect that what Weyl was saying was
that one man's variable is another man's name, which is indeed correct, but not
particularly deep or mysterious. (020)
>
> Well – a lot of people think they know what a variable is – and they are
>probably right – but how can something change (how can it vary) and still be
>the same thing? (021)
Easily. You change every minute of your life, every time you breathe, but you
are the same person. (022)
> This notion of identity is mysterious. (023)
I don't agree. It can be complicated, but its not *mysterious*. (024)
> In a computer, we might say that the variable or label is a box – that does
>not change over time – while the value “inside it” does change. Maybe
>this issue of “the box” and “what’s inside it” have something to do
>with the “figure/ground” mysteries that Richard Hofstadter was exploring
>– a kind of primal yin/yang.
>
> All human understanding is constructed on something like this basis.
>Alphabets come from somewhere, words are constructed from them, and from words
>come huge complex edifices – and then huge arguments erupt within those
>edifices – and the welfare of the entire human race is put at risk.
>
> My instinct has been – to drive the analysis of this structure (025)
Which structure, exactly? Are you talking about the syntax of language? (026)
> down to an absolutely immovable (undifferentiable) foundation – probably
>involving concepts like the “Dedekind cut” that partitions the real number
>line and defines the principle of continuity. (027)
Whoa. What has syntax (or indeed semiotics) got to do with the real number
line? Your needle skipped a few grooves at this point. (BTW, the definition of
Dedekind cut  or indeed any other way to define the continuum, as opposed to
the mere rational line  is inherently secondorder, so cannot be fully
expressed in any firstorder system, and cannot be fully captured by any
computable operations upon any syntax. All this is standard textbook stuff
about recursive computability and logical expressivity.) (028)
> Conceptual reality comes into the world at the point where
>(digital/finitestate) concepts parse or intersect with continuous (or
>undifferentiated) reality and start drawing arbitrary boundaries selected for
>human purposes. (029)
Actually, there is a very convincing argument that the real universe is
discrete rather than continuous, and finite (very, very large, but finite)
rather than infinite. See http://motionmountain.net/ , especially volume 4 et.
seq.. In which case, all this late19thcentury mathematics of the continuum
has no physical meaning at all. Schiller explores the consequences of this very
thoroughly in the last volume. (030)
>
> So – I would like to see some way that “axioms” could be constructed
>from a primal basis like this – where every facet of the construction was
>explicitly identified – and perhaps reproduced in a machine space – and
>absolutely mapped into continuity with zero error. (031)
I don't even know what it would mean to "map into continuity with zero error".
I stringly suspect that it doesn't mean anything :) (032)
> If we really could really get solid on this – we might have some hope of
>getting past this generallyaccepted convention that “even axioms are a
>matter of philosophic opinion” – and begin to unfold a true common ground
>and common foundation for the entire undertaking of civilization… (033)
Before trying to build a new philosophy of the nature of axioms, why don't you
first become familiar with the existing ways of using axioms? Then at least you
will be in a better position to criticize them and to build something better.
Standing on the shoulders of giants, and so forth. (034)
Pat Hayes (035)
>
> ???
>
>  Bruce
>
>
>
> <image002.png>
>
>
> <image001.png>
> <image003.png>
>
>
> Peano axioms
> From Wikipedia, the free encyclopedia
> In mathematical logic, the Peano axioms, also known as the Dedekind–Peano
>axioms or the Peano postulates, are a set of axioms for the natural numbers
>presented by the 19th century Italian mathematician Giuseppe Peano. These
>axioms have been used nearly unchanged in a number of metamathematical
>investigations, including research into fundamental questions of consistency
>and completeness of number theory.
>
> The need for formalism in arithmetic was not well appreciated until the work
>of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic
>could be derived from more basic facts about the successor operation and
>induction.[1] In 1881, Charles Sanders Peirce provided an axiomatization of
>naturalnumber arithmetic.[2] In 1888, Richard Dedekindproposed a collection
>of axioms about the numbers, and in 1889 Peano published a more precisely
>formulated version of them as a collection of axioms in his book, The
>principles of arithmetic presented by a new method (Latin: Arithmetices
>principia, nova methodo exposita).
>
> The Peano axioms contain three types of statements. The first axiom asserts
>the existence of at least one member of the set "number". The next four are
>general statements about equality; in modern treatments these are often not
>taken as part of the Peano axioms, but rather as axioms of the "underlying
>logic".[3] The next three axioms are firstorder statements about natural
>numbers expressing the fundamental properties of the successor operation. The
>ninth, final axiom is a second orderstatement of the principle of mathematical
>induction over the natural numbers. A weaker firstorder system called Peano
>arithmetic is obtained by explicitly adding the addition and multiplication
>operation symbols and replacing the secondorder induction axiom with a
>firstorder axiom schema.
>
> **
>
> An axiom, or postulate, is a premise or starting point of reasoning. A
>selfevident principle or one that is accepted as true without proof as the
>basis for argument; a postulate. As classically conceived, an axiom is a
>premise so evident as to be accepted as true without controversy.[1] The word
>comes from the Greek ἀξίωμα (āxīoma) 'that which is thought worthy or
>fit' or 'that which commends itself as evident.'[2][3] As used in modern
>logic, an axiom is simply a premise or starting point for reasoning.[4] Axioms
>define and delimit the realm of analysis; the relative truth of an axiom is
>taken for granted within the particular domain of analysis, and serves as a
>starting point for deducing and inferring other relative truths. No explicit
>view regarding the absolute truth of axioms is ever taken in the context of
>modern mathematics, as such a thing is considered to be an irrelevant and
>impossible contradiction in terms.
>
> In mathematics, the term axiom is used in two related but distinguishable
>senses: "logical axioms" and "nonlogical axioms". Logical axioms are usually
>statements that are taken to be true within the system of logic they define
>(e.g., (A and B) implies A), while nonlogical axioms (e.g., a + b = b + a)
>are actually defining properties for the domain of a specific mathematical
>theory (such as arithmetic). When used in the latter sense, "axiom,"
>"postulate", and "assumption" may be used interchangeably. In general, a
>nonlogical axiom is not a selfevident truth, but rather a formal logical
>expression used in deduction to build a mathematical theory. As modern
>mathematics admits multiple, equally "true" systems of logic, precisely the
>same thing must be said for logical axioms  they both define and are specific
>to the particular system of logic that is being invoked. To axiomatize a
>system of knowledge is to show that its claims can be derived from a small,
>wellunderstood set of sentences (the axioms). There are typically multiple
>ways to axiomatize a given mathematical domain.
>
> In both senses, an axiom is any mathematical statement that serves as a
>starting point from which other statements are logically derived. Within the
>system they define, axioms (unless redundant) cannot be derived by principles
>of deduction, nor are they demonstrable by mathematical proofs, simply because
>they are starting points; there is nothing else from which they logically
>follow otherwise they would be classified as theorems. However, an axiom in
>one system may be a theorem in another, and vice versa.
>
>
>
> From: ontologforumbounces@xxxxxxxxxxxxxxxx
>[mailto:ontologforumbounces@xxxxxxxxxxxxxxxx] On Behalf Of Barkmeyer, Edward
>J
> Sent: Tuesday, April 22, 2014 9:24 AM
> To: [ontologforum] ; rhm@xxxxxxxxxxxxx
> Subject: Re: [ontologforum] axiom
>
> Bruce,
>
> The simple definition of ‘axiom’ is “a statement/sentence that you take
>to be true”, usually without having any ability to prove its truth from
>other knowledge or assumptions. There can be many different reasons WHY
>someone takes a given statement to be true – social agreement, blind faith,
>preponderance of evidence, hubris, etc. That does not affect the concept
>‘axiom’.
>
> There are other vernacular uses of the term ‘axiom’, but with respect to
>the discipline of ontology development, an axiom is just a sentence taken to
>be true.
>
> It is worthy of note that 99% of all ontologies, information models and UML
>models consist of nothing but terms and axioms. Everything captured in the
>model is offered without any evidence for its validity. The theorems – the
>statements that can be proved using the ontology – are the answers to the
>questions that form the use cases for the ontology.
>
> Ed
>
> From: ontologforumbounces@xxxxxxxxxxxxxxxx
>[mailto:ontologforumbounces@xxxxxxxxxxxxxxxx] On Behalf Of Bruce Schuman
> Sent: Monday, April 21, 2014 12:14 PM
> To: rhm@xxxxxxxxxxxxx; '[ontologforum] '
> Subject: Re: [ontologforum] axiom
>
> What is an axiom?
>
> For me – in this context of “semantic ontology” and computer science,
>where we are trying to be highly precise, these below definitions of
>“axiom” are like nonsequiturs. I don’t get this.
>
> They might refer to things that are “axiomatic” by social convention –
>but the way I see it, any general statement written in abstract terms in the
>English language cannot be unambiguous, and therefore cannot be
>“axiomatic”.
>
> Let’s say that the axiom we want to explore is “All dogs are free”. We
> might get some social agreement on that among certain groups.
>
> But for me – trying to build an inviolate irrefutable system based on that
>idea, that would lead to noncontroversial results – is just impossible, and
>based on a fundamental misunderstanding.
>
> What do these words mean?
>
> What is “a dog”?
>
> What is “free”?
>
> Maybe we know what “are” means – maybe we don’t.
>
> Maybe we know what “all” means (if we know what “a dog” is) – and
>maybe we don’t.
>
> All those terms and definitions are stipulative. “A dog is what I say it
>is”. “Freedom is what I say it is”. And that’s the only certain
>meaning those words have.
>
> And if we are a follower of some philosopher – maybe those words mean what
>that philosopher says they mean. But even that is not likely to produce
>stable results, as the followers are very likely to start quarreling about the
>specific implications. We need the philosopher/guru standing around to give
>us his/her authoritative “correct” answer (opinion) on any questions that
>arise. We’re not going to be realizing Leibniz’s dream: “Let us
>calculate…”
>
> Under any kind of “load stress” (critical examination or controversy), a
>proposition/axiom stated in those broad abstract terms – leads to
>crazymaking. It doesn’t work. Don’t do it.
>
> So – for me, scratching my head and wondering why so many messages have
>been posted on this subject – maybe I just don’t understand something that
>everybody else agrees on – or maybe this concept of “axiom” needs a
>hardedged definition in this “ontolog” context.
>
> If I had my way, every natural language term would be (stipulatively)
>grounded (somehow) in the real number line, with exact measurements in x
>number of decimal places in explicit dimensions with a known error tolerance..
> A “dog” is exactly this….
>
> That’s the only way to stop the arguments and lead to stable structures.
>
>
>
> Bruce Schuman
>
> From: ontologforumbounces@xxxxxxxxxxxxxxxx
>[mailto:ontologforumbounces@xxxxxxxxxxxxxxxx] On Behalf Of Richard H.
>McCullough
> Sent: Sunday, April 20, 2014 10:48 PM
> To: [ontologforum]
> Subject: [ontologforum] axiom
>
> The New Shorter Oxford English Dictionary gives three definitions.
>
> 1) a) An established or generally accepted principle;
> b) a maxim;
> c) a rule.
>
> 2) Logic. A proposition (true or false).
>
> 3) Math.
> a) A selfevident truth;
> b) a proposition on which an abstractly defined structure
> is based.
>
> Rand:axiom is 3a;
> McCullough:axiom is 3b;
> Ontolog Forum:axiom is ?;
>
>
> Dick McCullough
> Context Knowledge Systems
> Name your propositions !
>
>
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