[FOM] To Vladimir Sazonov and others doubting the unambiguityof N

Aatu Koskensilta aatu.koskensilta at xortec.fi
Wed Jun 25 03:49:02 EDT 2003

Vladimir Sazonov wrote:

> Again, I am sorry for the late reply. 
> Aatu Koskensilta wrote:
>>Vladimir Sazonov wrote:
>>>Aatu Koskensilta wrote:
>>>>Vladimir Sazonov wrote:
>>>>>What (as you say) "disinformed" me? Some deeper that in the school
>>>>>things like Goedel's theorems, especially on incompleteness and
>>>>>Goedel/Cohen proof on independence of CH, what demonstrated (to me)
>>>>>that both N and continuum are vague concepts, [ --- ]
>>>>Like [] I can understand your position with regards to the continuum,
>>>>but as to N, I'm still baffled. Surely the notion of *non*-standard
>>>>model of arithmetic is much more illusive, as any such model must
>>>>necessarily be non-recursive?
> Feasible numbers 0, 0', 0'', AND SO ON, 
> are a nonstandard model of (of course, not Peano) arithmetic. 
> But they are quite real, not illusive. 

So the feasibility of a number is determined solely on basis of the 
length of the numeral sequence denoting it. To me, most numnbers above 
10 would be nonfeasible; that is, I can't work with them at ease and 
often not at all. I have to resort to other means of understanding and 
calculation for these numbers, which means I believe you would classify 
as implicit usage of a formal systemns, such as PA.

I can't see how feasible numbers (let alone feasible-to-me numbers) 
could server as a model of arithmetic. Could you elaborate on what sense 
these are a model of arithmetic?

>>>Here I feel you assume ZFC or the like where what you mentioned
>>>makes sense. As I already wrote (actually many times), in this
>>>framework I have no problems with understanding the concept of
>>>standard or nonstandard models of PA. All of this is defined
>>>(or proved) in ZFC quite precisely. I have a serious problem
>>>with understanding when "standard" model of PA is mentioned
>>>in some ABSOLUTE, metaphysical, quasireligious sense (not
>>>RELATIVE to ZFC or the like).
>>This is exactly what I fail to comprehend. It seems that since we have
>>the metatheorem in systems like ZFC and the like about PA that all its
>>recursive models are isomorphic and *standard*, this shows that we
>>*can't*, without assuming quite strong platonistic framework even
>>produce any non-standard models for display, or to have any point of
> First, I cannot understand which way something proved in ZFC, even 
> about standard model of PA (quite a legal, INTERNAL object of ZFC), 
> is related with what you are asking. This seemingly means that you 
> assume some ABSOLUTE, EXTERNAL to ZFC standard N. 

There are absolute facts about our understanding of the natural numbers 
that render them liable to be models of certain theorems. This means 
only that our actual practice and all has such features (in 
idealisation) that certain formal results can be applied. The fact that 
these formal results are results in a formal theory is of no 
consequence; if it's the case that our intuitive notions (suitably 
modified) behave sufficiently similarly to some formal notions, we can 
use these formal notions to reason about our intuitive notions.

The fact that all non-standard models of arithmetic are non-recursive 
simply means, when applied to our actual pratice, that no constructive 
means can get us a non-standard model of arithmetic. If one believes in 
a form of mechanism according to which our reasoning is in some sense 
recursive, this would seem to imply that we needn't concern ourselves 
with such philosophical problems as whether two people talking about N 
are really talking about the same thing, since there is - unless one 
assumes certain non-constructive principles - no non-standard model 
there could be confusion about.

Perhaps one way to understand this is to consider the thesis of Hartry 
Field that mathematics is always a *conservative* extension of the 
theory in which it is incorporated (provided the theory itself is 
sufficiently interesting). The result that all recursive models of PA 
are isomorphic can then be taken to be a mere restatement of the fact 
that two people calculating things with recursive functions and talking 
about natural numbers could never disagree (correctly). There is no need 
to buy into all the metaphysical baggage of the theory in which this 
result is proved, it is sufficient to assume the soundness of this 
theory with respect to some restricted class of statements.

> On the other hand, for BA = Bounded Arithmetic which I mentioned in
> another posting, take BA + ~EXP. Forget about existence of ZFC with 
> all its model theory, and imagine a "model" of this theory with
> exponential *partial* recursive function. Quite reasonable, intuitive 
> theory which even better corresponds to our real world. This 
> imaginary model is quite different from any imaginary model of PA 
> and can be called, by this reason, non-standard. No strong 
> platonistic framework is assumed. 

But surely it's not a model of PA? I agree that such a system is very 
interesting and merits study. (I snipped some of your points because 
while they're interesting I'm not familiar enough with the subject to 
say anything about them).

> But the main problem consists in asserting some absolutely 
> unnecessary super beliefs which are coming rather from philosophy 
> of mathematics rather than from mathematics itself. 

The "belief" in standard N is not coming from philosophy. Rather, it's 
coming directly from mathematics. I find it strange that I should 
suddenly start doubting all sorts of mathematically obvious things just 
because I'm doing logic. Similarly, I believe it would be hypocritical 
of me if I did the same when doing philosophy. I don't think I can 
rightfully go about doing mathematics and at the sime time saying things 
like "N is vague" or the like.

> Let us recall, whether non-Euclidean Geometry "manifested itself" 
> before it was discovered? Just vice versa, there were philosophical 
> views just against this. Just Euclidean Geometry was considered as 
> the only possible in principle, some ABSOLUTE. This was a serious 
> problem for Gauss to publish his results. Lobachevsky, who dared 
> to publish his Imaginary Geometry was considered crazy. 

Are you saying that there could be non-Dedekindian (or non-Peanoan) 
arithmetic we're suppressing with the assumption of a "standard" N?

> Recall also the absolute space-time. If the eyes of scientists 
> would be opened for potential possibilities, the Relativity Theory 
> could arise earlier, first as an imaginary possibility. Just 
> ask the question what does it mean the absolute space-time. 
> which exactly experiments confirm that there is the unique time 
> for distant physical objects. After asking such "stupid" questions 
> and finding appropriate experiments we immediately come to the 
> alternative possibility, just in principle because the experiments 
> could, in principle, give different results. 

I haven't seen you offer even a picture of what it would mean for some 
arithmetical statement to be "undetermined" in the same sense as some 
geometrical statements (which aren't results of absolute geometry). I 
simply can't even imagine what this would be like.

> Not asking such questions means relying on some beliefs, like 
> the existence of a God who is the carrier of the absolute time. 
> Which other way it could be absolute. Only signals with an 
> infinite speed could be the reason for the absolute time. 
> But the infinite speed is something suspicious. Equally 
> suspicious for me are absolute truth or absolute, standard N. 
> Should we learn something from the history of Science and, 
> in our case, from what was happened with CH? 

There are relevant differences here. First of all, we are *much* more 
familar with N than with any standard model of ZFC. In fact, we can get 
a model of N using only algorithms which are describable to us, these 
descriptions enabling us to carry out any construction within N. In 
Kreisel's terminology, it's the h-effectiveness of the notion of 
"standard" (there is no need for this prefix, for there is but one) N 
that makes it different from the intended models of set theory and such 
theories. This is a genuine epistemological distinction.

> When you work in ZFC, you seemingly consider just some imaginary 
> universe of sets. Which serious reason (besides any your beliefs) 
> does not allow you to think the same way on N and PA? 

None what-so-ever as regards to PA. As to N, it's not a formal theory 
but a mathematical object or structure. And by N I mean the structure 
one gets essentially by Dedekindian analysis; start with an element 0 
and a successor function and require the recursion theorem to hold. If I 
wished to interprete this not within mathematics, but within actual 
human practice, I would say that this result simply says that as long as 
the certain subset of our actions conforms to this normative idea, we 
can't get any disagreement.

> I don't think we need to relativise these notions to any
>>specific metatheory, even though the result itself is proved in some
>>You think that N is in some sense vague. How does this vagueness
>>manifest itself? Are there some theorems of arithmetic that don't have a
>>specific truth value? Are there some identities that aren't "decided"? I
>>can't see how N can be vague *unless* one assumes quite strong a theory,
>>in which it's supposed to be so (say ZFC).
> Of course, if we would find an undecidable hypothesis (may be P=/=NP 
> is such one) for which it will be demonstrated, as for CH in ZFC, 
> its independence of PA and even of ZFC and of many reasonable 
> extensions of ZFC, then we will have some demonstration of the 
> vagueness of PA which would convince many people. Is such a possibility 
> absolutely excluded? What is PA is in fact inconsistent? What then 
> would happen with this "standard" N you believe in? 

I refuse to speculate on such scenario. If PA turned inconsistent I 
would have witnessed for the first time in my life a mathematical truth 
refuted (PA being sound with respect to the intended semantics). The 
normative idea that mathematical truths can not be refuted is one of the 
notions that I hold absolute. Whatever would result from a contradiction 
in PA is something I can't comment; I can't even really imagine what it 
would be like. If PA turns out inconsistent, all bets are off. Depending 
on the exact nature of the contradiction, most of our mathematical 
knowledge turns out to be incorrect or at least unsubstantiated. I can't 
give any predictions as to what sort of modifications to our thinking 
and understanding this would lead to.

> I believe that any super beliefs are harmful. There is no rational 
> reasons for them. We will loose nothing if we reject these beliefs. 

Oh, but I believe there is very good reason to believe that N is in no 
sense vague.

>>Supposedly recursiveness is an absolute
>>epistemological notion. Do you agree with this? 
>>Or do you think that
>>recursiveness itself is in some sense relative to a formal theory?
> Yes. I demonstrated above how this is possible. 
> Anyway, this is one of mathematical notions and 
> can be considered ONLY in a framework of a formal theory. 
> Otherwise, what could you ever say about this notion 
> outside a formal (rigorous) reasoning? Trying to do that 
> is only self-deception. 

I don't see how mathematics could have any sensible content when 
interpreted according to your view. In your view we are implicitly using 
all these formal theories. But supposedly they have applications. And if 
they have applications, then this can only mean that portions of our 
actual practice or the world have such features as allow one to 
interprete them as models of the formal theory, and consequently use the 
formal theory as a tool for reasoning about these things. Now, 
recursiveness might be one of the notions that are formalised, and 
*applied* to actual real life world, in order to arrive at 
philosophically interesting results. If you deny this sort of 
applicability, then I'd be interested to know why you consider 
mathematics interesting in the first place -- there seems to be no 
inherent interest to arbitrary formal systems in themselves.

>>What you write is misunderstood or just ignored because it's alien, at
>>least to me (insert smiley here). 
> What is so alien? If you understand the universe of ZFC as something 
> not absolute, what is the problem to you to understand N in the same 
> way? 

The fact that unlike with ZFC I see no leeway for the vague wiggling 
abouts of the model to happen. The model is the recursive model, period.

> At first I thought you represented
>>some sort of Hilbertian formalsim (an eminently sensible view, by the
>>way), but your views now seem much more radical.
> It seems the only difference is that I pay attention to the fact 
> that derivations in a formal system are of feasible length. 
> Although this looks a trivial note, other people have a problem 
> with this because of knowledge of metamathematics where derivations 
> can be just imaginary ones. They identify imaginary with real 
> (what actually was noticed by Dana Scott as infinite regress). 
> Additionally, many want to have something absolute, if not the 
> universe of sets, then, at least, N. But why it is considered 
> so important??? 

Mathematical concepts are very often idealisations of actual practical 
concepts. For example, one could say that the concept of "natural 
number" is an idealisation based on our working with what you call 
feasible numbers. I just don't see the importance of rejecting these 
idealisations, as long as they serve our understanding. Similarly, when 
doing, say, computer program verification one usually does not assume 
some specific limited amount of memory.

>>>>There seems to be no such clear distinction with standard and
>>>>non-standard models of set theory (let alone the notion of "the"
>>>>standard model of set theory), and thus I can appreciate the idea that
>>>>there is something inherently vague to the continuum or the even more
>>>>substantially infinitistic set theoretic objects. But N? There seems to
>>>>be a genuine *mathematical* distinction here; the standard model is the
>>>>recursive model, and the non-standard ones are the non-recursive ones.
> I could only repeat that this is an internal fact of ZFC. 

Yes. But presumably the reason we use and study ZFC at all is that it 
has sound interpretations (at least with respect to certain class of 
statements). One of these interpretations gives way to the result that 
while working within recursive mathematics we can't come to a 
"non-standard" model of arithmetic.

> Externally, N considered in ZFC is just an imaginary object. 
> Externally, we have no rational way even to formulate what 
> does it mean that N is standard one, and there is nothing 
> to do as to consider N as imaginary one. How to compare 
> imaginations of various people whether they are the same? 
> We can compare only formal systems which we are using. 

The "imaginations" of two people are the same for mathematical purposes 
whenever they're "elementarily equivalent" (with respect to a suitably 
expressive language; certainly not of FOL!); when there is no (and 
modally, can be no) disagreement as to what is true of the 
"imagination". The recursiveness of the standard model of arithmetic 
implies that if we assume that these "imaginations" are recurzive, there 
can be no vagueness or misunderstanding as to N. Even if we contest the 
meaningfulness of the theories in which the recursiveness result is 
obtained, we must believe that if we can suitably interprete the formal 
theories "to be about" our imaginary constructs, and the theories come 
out true under these interpretations (not necessarily in the 
mathematical sense, perhaps only in an empirical or philosophical sense) 
then we must also accept the recursiveness theorem (as interpreted).

>>>The fact that WITHIN ZFC we have these results has no relation to
>>>my question on what is the ABSOLUTE standard N. I strongly believe
>>>that this is actually a wrong, fictitious concept having nothing
>>>rational behind of it. Usually mentioned abstractions of potential
>>>or actual infinity with respect to the ABSOLUTE standard N are
>>>themselves very vague.
>>But the absolute standard N only becomes vague when you wander outside
>>the realm of intuitionistic/constructionistic/finitistic mathematics!
> WHY??? How is it ever possible to define what is standard N in the 
> finitistic theory PRA? This is possible only in ZFC or the like!
> What did you mean at all in the above phrase? 

I meant that to get the idea that PA does not uniquely determine a 
specific mathematical structure, one must use it's incompleteness 
togerther with the completness theorem. There is no way to get a 
non-standard model of PA with 
intuitionistic/constructionistic/finitistic principles. Thus the model 
theoretic argument (which I take was one of the things that motivated 
your transition to your current position) presupposes the Platonistic 
framework (or at least a subset thereof) it's supposed to criticize.

>>This is very much unlike the case with ZFC and its standard and
>>non-standard models, let alone its "the standard model". You seem to see
>>no essential difference here, which baffles me.
> I do not understand which difference (which I "do not understand") 
> do you mean. 

There is no clear sense in which we can distinguish the standard model 
of set theory (whatever that means) from any other model in an 
epistemologically relevant sense. There *is* an epistemologically 
interesting sense in which N can be distinguished from evil non-standard 

>>>Say, potential infinity of N essentially assumes that we can
>>>always add 1 to any number. Moreover, it is assumed that we can
>>>ARBITRARILY iterate this our ability. It is this ARBITRARILY what
>>>is unclear for me. I understand that this assumes that by iterating
>>>the operation x+1 we should have the ability to always fulfill the
>>>operation x+y. Further iterations lead to multiplication,
>>>exponential, superexponential operations, primitive recursive
>>>functions, Ackermann's function,..., epsilon-0-recursive functions,
>>>                      AND SO ON.
>>>Yes, we can continue further and further, but how further?
>>>Until we will get tired? What this AND SO ON really means?
>>>Can anybody explain? If not, then this is something indefinite,
>>>vague. Thus, the "resulting" N is also vague. Let us be honest
>>>before ourselves.
>>I don't see why this needs to be vague at all. It is true that many
>>philosophers in the past have formulated this "going on arbitrarily" in
>>a misinformed fashion, even claiming that we are somehow "compulsed"
>>mentally to always go on. This compulsion exists, but it is of modal
>>nature; we *can* always go on, and this continuation is unambiguous
>>*provided* one doesn't actually work in some strong platonistic
> Just vice versa, this may be done unambiguous only when we 
> formalize (explicate) this process in some way. This may be done 
> only in (or relative to) a sufficiently strong (meta)theory. 

Why? If the only clarity we can achieve is trough formal systems, it 
seems that we can have no clear understanding of the formal systems, at 
least not without working within another formal system. There seems to 
be a problem in connecting these formal systems to reality in a 
meaningful way, an infinite regress of formal theories.

> The so-on simply means that we *can* always consider the
>>natural numbers further ahead, and that they are uniquely determined by
>>the production rule (and the fact that whatever recursive function we
>>might consider, it acts "correct" for these numbers, i.e. in effect
>>Dedekind's recursion theorem).
> How can you discuss all of this outside a formalism? 

Easily. I just did.

> I also tried to discuss above the potential infinity, 
> outside a formalism, but I honestly said that I am unable 
> to do this in a definite way. I do not know what does it mean 
> this "and so on" in general. You seems assert that you know,  
> but by which miracle? 

Well, here we come to a point where there simply seems to be nothing to 
argue about. It is obvious to me what it means that the succession of 
natural numbers does not end. It's apparently not obvious to you. Unless 
we can get to a deeper rooted disagreement, there's not much either of 
us can do to convince the other.

>>>Again, what is the "length" of the resulting N? Intuitively,
>>>it is much more comfortable for me to think (together with
>>>Esenin-Volpin) about many (infinite) Ns of various "length",
>>>with various abilities to iterate the ability to iterate the
>>>operation x+1. It is intuitively plausible that the simple
>>>iteration of x+1 leads us only to feasible numbers where
>>>2^1000 is non-feasible.
>>I can't understand how any number we can actually name and work with
>>could be non-feasible!
> Is this the question on the (very informal) definition I already 
> presented and explained in detail, or only on the terminology? 
> Will you be precise, please. 

I see now that my question is not actually relevant given your usage of 
feasible number as one we could actually write down in "unary" notation 
(or some such system).

> I also wrote that natural numbers, like this one, are quite 
> legal objects of PA. Why should I repeat? 

You need not repeat. I understand that you consider manipulations of 
such numbers a 2^100000000213 to take place within a formal theory of 
arithmetic - at least implicitly (take place implicitly, that is). I 
don't see why we can't reason about such numbers without resort to 
formal systems, and I can't see what fruitful consequences can one draw 
from the assumption that underlying all such actions lies a formal theory.

>>>Of course, we can, in principle, make this AND SO ON explicitly
>>>defined WITHIN a formal theory. But this will mean that we
>>>relativized this AND SO ON and corresponding version of N to a
>>>formal theory. (QED!) In general, the only possibility to do
>>>something precise in mathematics is via formalizing.
>>Your conclusion only follows if one accepts your thesis that the so-on
>>is somehow inherently vague. I don't see why this needs to be so.
> This is probably because you are too religious concerning N. 
> Any rational questions concerning the "holy" N are forbidden? 

No. I just don't think your questioning is in any sense valid (or if it 
is, it can be answered). I'm not trying to prevent you from doing 
anything. You are free to explore whatever thoughts you wish.

>>>>From your postings I gather this won't satisfy you, but I'd be
>>>>interested to know whether Gödel's theorems merely motivated you to
>>>>question the platonistic picture of mathematics or do you believe they
>>>>server as arguments against such a position?
>>>As I wrote in a posting to FOM, they are (may be indirect) witnesses
>>>of the vagueness of N. They stimulated me to start doubting and asking
>>>the question "what is the standard model of PA?". I have no direct
>>>answer, and, I believe, nobody has. Why then to use this "wrong"
>>>concept (except explicitly within ZFC) at all?
>>There is a kernel of truth in what you say. I believe we don't need a
>>concept of "the standard model of PA" unless we're already working in a
>>strong theory, 
> looks like you understood me, but...
>>simply because there's no way for non standard models to
> No, simply because there is no rational way to explain what it is, 
> like there was no rational way to explain why the only possibility 
> for the geometry is to be Euclidean. 

There never was a mathematical proof that all geometries must be 
Euclidean. There is a mathematical proof that all models of arithmetic 
that are recursive (i.e. the only models we can talk about without 
assuming quite substantial Platonistic principles) are isomorphic. This 
result is a result in many formal systems, but the reason I take it to 
be relevant is that I believe that (certain idealized versions of) our 
mathematical practices can be interpreted in light of this theory. And 
if they can be interpreted, any theorems of the theory must also hold 
(when interpreted similarly).

The only reason I brought up the recursiveness of the standard model was 
because you said that Gödel's incompleteness theorems motivated you to 
adopt your current philosophical position. I think that the 
recursiveness result shows that such motivation is ill-guided - quite 
similar to the problems one has with interpreting portions of a 
statement intuitionistically and other portions classically and then 
wondering why it makes no sense.

You seem to have independent reasons to belive in the vagueness of N, so 
this is not an all important point. However, if your only reasons are 
that you can't understand "how to go on", then I can only say that I 
don't believe it. Perhaps you mean that you don't know that in any 
situation you would know how to go on, then I could better understand 
you (but still disagree).

> You seems want to have any belief, if there is nothing (from your 
> point of view) against it. 

No. I want to have a belief if there is a very strong reason for it; the 
recursiveness of the standard model.

> I, as soon as I able to, avoid any beliefs in mathematics and 
> science. There are quite rational theories which replace them. 
> Down with irrational super beliefs in Mathematics! 
> They are not only useless and meaningless, but potentially harmful, 
> like the beliefs in the unique geometry or in absolute space time. 
> This view is much more healthy. 

If all you have is theories without any clear connection to any prior 
"beliefs", how do you decide which theories are interesting and worth 
studying? Or what (classes of) statements should be studied within such 

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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