What is the nature of the axioms of Objectivism?

[dougclayton]

I got an error message from your first link. Your second link provides discussion that shows the priniciple isn't reduced to saying that the simplest explanation is correct. To say that one should not make unneeded assumptions is just not equivalent to saying that the simplest explanation is correct. As to being an axiom, I haven't opined that it should be one.

<snip extended discussion of how "my formulation" is silly>

(emphasis mine)

I'm not sure what your intent is here, unless it is to set me straight on a pet subject of yours. Given that the original poster offhandedly posited Occam's Razor as an axiom, and I summarized it informally to show it wasn't, and you haven't disagreed with that analysis, I wonder where all this objection is coming from. Did I err in concluding it wasn't an axiom? Did the difference between how I summarized it and how you do affect that analysis at all? This isn't a general discussion on the utility of Occam's Razor, nor is it a philosophical treatise on all its implications, nor have I attempted to claim there was anything wrong with it--only that it is not an axiom.

So unless you want to tell me how I arrived at the wrong conclusion regarding it being an axiom--which I would definitely like to hear--I have to say your objection is just nitpicking. Let me state for the record that you are superior to me in wisdom regarding Occam's Razor, and I will hereinafter quote it exactly as "Entities should not be multiplied beyond necessity" so as not to make any more silly formulations.

[DavidOdden]

You quoted a version of Occam's Razor, fair enough. And you stated a principle that statements should not be without positive or negative evidence, fair enough. But you also claim that the quote and your own statement convey the same principle. I just said that that is for you to demonstrate or not, as you choose. That doesn't imply that I hold that the quoted version of Occam's Razor is not a reasonable summary of the principle. There's no ball in my court here.

Fine, you have no balls there. I don't understand what you think is left for me to 'demonstrate'. You're just engaging in automatic gainsaying. If you reject OR, you can just say so and be done with it. If you don't understand it, you can say so. Otherwise, I don't see what you could possible question.

[LauricAcid]

<snip extended discussion of how "my formulation" is silly>

My point (not given extended discussion, by the way) was not that your formulation was silly, but rather that if we were to accept that your formulation is an accurate rendering of Occam's Razor then this would make Occam's Razor look silly. In other words, your formulation was a strawman, even if unintentionally. But I do appreciate (no sarcasm or condescension meant) that you recognize now that your formulation was incorrect.

I'm not sure what your intent is here, unless it is to set me straight on a pet subject of yours.

Occam's Razor is not a special area of interest for me. My intent was just to note that your formulation should not be taken as a formulation of Occam's Razor.

nor have I attempted to claim there was anything wrong with [Occam's Razor] --only that it is not an axiom.

Actually, you argued that the principle has a weaknes: "Why can't the complex explanation be true? How do I rely on the fact that the simplest explanation is true when I claim that a complex explanation is true?" And that was enabled by your incorrect formulation of the principle, since the principle does not preclude that a more complex explanation may be a better one.

So unless you want to tell me how I arrived at the wrong conclusion regarding it being an axiom--which I would definitely like to hear--I have to say your objection is just nitpicking.

Even if your error was not related to your conclusion, it is still worthwhile to note that your formulation was not that of Occam's Razor.

Let me state for the record that you are superior to me in wisdom regarding Occam's Razor

And let me state for the record that I am not and do not claim to be an authority on the subject of Occam's Razor. Edited September 24, 2005 by LauricAcid

[LauricAcid]

Fine, you have no balls there.

Ouch.

I don't understand what you think is left for me to 'demonstrate'.

You claim that the principle of Occam' Razor (OR) and your principle that statements should not be without positive or negative evidence (I'll abreviate this as 'SE') are the the same principle. So, that OR and SE are the same principle is what is is left for you to demonstrate. Actually, since it is your principle that statements should not be without positive or negative evidence, then you could put that principle into action here by providing the positive evidence that that principle is the same principle as Occam's Razor.

You're just engaging in automatic gainsaying.

No I'm not (or was that also "automatic gainsaying"?). You made an assertion that SE is the same as OR. I just asked: What is the basis for your assertion?

If you reject OR, you can just say so and be done with it.

My asking why you think SE and OR are the same principle does not depend on my rejecting or accepting OR.

If you don't understand it, you can say so.

OR leads to some deep issues in philosophy, and I surely don't claim to have a complete understanding of all of them. But in this context I was just aksing why you think SE is the same principle as OR.

Otherwise, I don't see what you could possible question.

You don't see how one can possibly question your assertion that SE is the same as OR? Are you now claiming that not only are SE and OR the same principle but that it is self-evident that they are the same principle? Edited September 24, 2005 by LauricAcid

[DavidOdden]

You don't see how one can possibly question your assertion that SE is the same as OR? Are you now claiming that not only are SE and OR the same principle but that it is self-evident that they are the same principle?

I don't think that it's immediately and perceptually self-evident, but it's close enough. By providing the actual statement, I've make a good case for my position, although if you understood Peikoff's account of certainty, you could argue that the conclusion is not certain. Do you at least grant that OR makes a normative claim, and that the injunction is suspended when it is "necessary"? If so, then your only rational basis for doubt would be that you have some evidence for a competing account of what is "necessary". I just haven't seen what that evidence is or even what that competing account would be.

[LauricAcid]

By providing the actual statement, I've make a good case for my position, although if you understood Peikoff's account of certainty, you could argue that the conclusion is not certain.

What actual statement do you mean? What conclusion? The conclusion that OR and SE are the same principle?

Do you at least grant that OR makes a normative claim, and that the injunction is suspended when it is "necessary"?

I don't know what you mean by the injunction (OR presumably) being necessary in some instances as opposed to not necessary in other instances.

If so, then your only rational basis for doubt would be that you have some evidence for a competing account of what is "necessary".

What I've doubted is that OR and SE are the same principle. I just don't know what you have in mind with the notion of necessary here.

[DavidOdden]

What actual statement do you mean?

Pluralitas non est ponenda sine neccesitate.

I don't know what you mean by the injunction (OR presumably) being necessary in some instances as opposed to not necessary in other instances.

That's what the sine is all about.

I just don't know what you have in mind with the notion of necessary here.

Okay, I see the problem. You don't understand the concept "necessary". Maybe I can dig up some good references that will explain the concept for you.

[LauricAcid]

The question I asked was why you think OR and SE are the same. Now I understand that you say that by quoting OR you've made a good case for your position. What good case? What case at all? Just holding up two sentences side by side and claiming them to be equivalent is not an argument that they are.

As to 'necessary', now I see that your 'it' was not to the antecdent 'injunction'. Fair enough. So my answer is that of course OR should not be understood to advocate eliminating assumptions that are needed for the explanatory role of a theory.

As to my understanding the concept of necessary, I am familiar with many facets of the concept. However, your own definition of it and explanation of your understanding of it are welcome.

Edited September 24, 2005 by LauricAcid

[robstercraws]

Umm, back to the main topic of Objectivist axioms. I just want to clarify a few things...

1. The statement "they (axioms) can be verified directly through sense perception" is wrong. Axioms by definition are accepted to be true. To try to verify an axiom means you don't understand the meaning of axiom. Again they're true by definition not verification.

2. Also be aware that in these "axioms" are axioms in only the most general sense (as assumed truths for purposes of argument) but aren't logic axioms as defined in formal logic. These Objectist "axioms"don't display the propery of being axiomatic or atomic. Meaning they're comprised of multiple ideas allowing for abiguity and implied meaning. Look at Peano's axioms of mathematics if you want to see true axioms. These are true logical axioms because they can not be broken down in any simpler ideas.

[DavidOdden]

1. The statement "they (axioms) can be verified directly through sense perception" is wrong. Axioms by definition are accepted to be true. To try to verify an axiom means you don't understand the meaning of axiom. Again they're true by definition not verification.

This is a surprisingly common misconception. Axioms are not "true by definition", they are simply "true". It isn't a matter of arbitrary acceptance. It is also not necessary to verify the truth of an axiom, but also certainly entirely possible. An axiom is a self-evident truth, but the concept of self-evidence requires perception.

2. Also be aware that in these "axioms" are axioms in only the most general sense (as assumed truths for purposes of argument) but aren't logic axioms as defined in formal logic. These Objectist "axioms"don't display the propery of being axiomatic or atomic.

On the contrary, the Objectivist are as axiomatic and atomic as you can get in philosophy. Peanos axioms are highly derivative, meaning that they depend on many lower-order concepts. For example it is not axiomatic that there are natural numbers and numbers that are not natural; the concept of "successor" is certainly not axiomatic, nor is it axiomatic that 0 is a natural number (though it might be a derivative conclusion, depending on how "number" and "natural" are defined. Indeed, -1 has the successor 0, so either the Peano Axioms are false, or "natural number" is a non-self evident term of art.

[AisA]

Umm, back to the main topic of Objectivist axioms. I just want to clarify a few things...

1. The statement "they (axioms) can be verified directly through sense perception" is wrong. Axioms by definition are accepted to be true. To try to verify an axiom means you don't understand the meaning of axiom. Again they're true by definition not verification.

Why? Because you say so? This sounds like the "axioms are arbitrary" argument.

2. Also be aware that in these "axioms" are axioms in only the most general sense (as assumed truths for purposes of argument)

Objectivism's axioms are not assumptions. They are perceptually self-evident. Are we to understand that you do not consider the fact of existence and your consciousness of it to be self-evident? If not, we have nothing else to discuss.

but aren't logic axioms as defined in formal logic. These Objectist "axioms"don't display the propery of being axiomatic or atomic. Meaning they're comprised of multiple ideas allowing for abiguity and implied meaning.

Where is the ambiguity and multiple implied meaning in "Existence exists"?

Look at Peano's axioms of mathematics if you want to see true axioms. These are true logical axioms because they can not be broken down in any simpler ideas.

The Peano axiom that I remember is "Zero is a number". That statement most certainly presupposes a number of more fundamental ideas, among them existence, identity and consciousness.

[LauricAcid]

Axioms by definition are accepted to be true. To try to verify an axiom means you don't understand the meaning of axiom. Again they're true by definition not verification.

There are different definitions and understandings of what axioms are, through mathematics, logic, and philosophy; and Objectivism has its own concept of axioms. It is pretty much senseless to talk about all of these at once. For example, in modern formal axiomatics there are two kinds of axioms: logical and non-logical. Logical axioms are logically true while non-logical axioms are true in some models but not in others. So, even in formal axiomatics it is incorrect to say that axioms are assumed to be true without qualifying as to logical and non-logical axioms and without qualifying that non-logical axioms are true only in certain models.

Also be aware that in these [Objectivist] "axioms" are axioms in only the most general sense (as assumed truths for purposes of argument) but aren't logic axioms as defined in formal logic.

They are not axioms of a formal system, but Objectivists argue that the axioms are not just assumed to be true but rather are true in the sense that to deny their truth would be self-contradictory.

These Objectist "axioms"don't display the propery of being axiomatic or atomic.

In the study of formal systems, the word 'atomic' has a special meaning and it is not required that axioms be atomic.

Meaning they're comprised of multiple ideas allowing for abiguity and implied meaning.

The reason for any ambiguity is just that they're not in a formal language. But even in a formal language, having multiple "ideas" does not disqualify a formula from being an axiom. Morevover, even formal axioms imply meaning as they narrow the class of models of the theory. And, the general notion of axiom is not limited to formal languages anyway. Yet, in another sense, even formal axioms are "ambiguous" unless the theory has only isomorphic models.

Look at Peano's axioms of mathematics if you want to see true axioms. These are true logical axioms because they can not be broken down in any simpler ideas.

(1) If the Peano axioms are put in a formal language, then they're formal axioms. But there is no requirement that formal axioms cannot use defined terms or that they not be derivable from other axioms. It is true that usually mathematicians greatly prefer to have axioms that are not derivable from one another, but this is not a requirement. (2) You're conflationg primitives with axioms. The primitives are the operation symbols and predicate symbols that other operation and predicate symbols are defined from, but axioms are not operation symbols or predicate symbols (except one could possibly take a 0-place predicate symbol as an axiom) but rather they are (well formed) formulas. (3) The Peano axioms, even in a formal system, are not logical axioms since they are true in some models and not in others.

EDIT: Even logical axioms of first order logic are not just assumed to be logically true, but are proven to be.

Edited September 26, 2005 by LauricAcid

[LauricAcid]

Peanos axioms are highly derivative, meaning that they depend on many lower-order concepts.

Axioms may be derivable; while operation and predicate symbols may be definable. And both derivability and definability are always relative to a particular formal system. If the Peano axioms are in first order logic (in which case the induction axiom is an axiom schema), then, of course, the Peano axioms depend on the logical system itself, but the Peano axioms are independent (none is derivable relative to the system) as are the primitives (none are definable relative to the system) mentioned in the Peano axioms.

For example it is not axiomatic that there are natural numbers and numbers that are not natural; the concept of "successor" is certainly not axiomatic, nor is it axiomatic that 0 is a natural number

They are axiomatic relative to the particular theory. Also, the Peano axioms do not declare that there are numbers that are not natural numbers, but rather the axioms just don't mention one way or the other. It turns out that there are models of the Peano axioms that have objects that are not natural numbers, but this is a result of model theory, not something that can be stated within Peano theory itself.

(though it might be a derivative conclusion, depending on how "number" and "natural" are defined.

'natural number' can be defined in a theory "more basic" than Peano arithmetic, such as set theory. But mathematics takes 'natural number' as one predicate not as a predicate made from two predicates: 'natural' and 'number'.

Indeed, -1 has the successor 0

That depends on the definition of 'successor'. -1 is not in the theory of Peano arithmetic anyway, and even in the larger theory of set theory, -1 precedes 0 in the standard ordering of the integers, but 0 is not result of the set theoretic successor operation applied to -1.

so either the Peano Axioms are false, or "natural number" is a non-self evident term of art.

Not that that assertion follows logically from your incorrect explanation, but it is true that the Peano axioms are false in some models, while 'self-evidency', even if it applied to a predicate rather than a statement, is not at issue, since 'natural number' is primitive relative to Peano arithmetic and defined relative to set theory. Edited September 26, 2005 by LauricAcid

[DavidOdden]

Axioms may be derivable

and thus not axiomatic, in the general sense. I don't understand why they are called "axioms"; the term "postulate" is less likely to result in confusion since the Peano postulates aren't self-evident truths -- that are systemic stipulations that are the foundation of a particular kind of "result" (i.e. number theory).

They are axiomatic relative to the particular theory

In other words, they are not part of an overall integrated epistemology, but are based on the false assumption that you can extract certain gobbets of knowledge and treat them in isolation from other gobbets of knowledge, and when you discover that you really cannot construct these well-defined boundaries that result in "pure mathematics" with no reference of real-world entities, then you have to introduce pseudo-axiomatic statements so as to avoid facing the connections between this aspect of knowledge and that aspect of knowledge. Still, that doesn't make such postulates axiomatic.

-1 is not in the theory of Peano arithmetic anyway, and even in the larger theory of set theory, -1 precedes 0 in the standard ordering of the integers, but 0 is not result of the set theoretic successor operation applied to -1.

That is certainly not self-evident; in fact is seems downright arbitrary. It is obvious that 0 is the successor of -1 just as 3 is the successor of 2. I assume then that there is just this arbitrary stipulation for come kinds of mathematician that the term "sucesssor" is undefined for the predecessor of zero. Whatever. I'm not saying you can't make these kinds of arbitrary claims, just that it is not self-evident and therefore not axiomatic.

Getting this back to the main topic, arbitrary stipulations aren't axiomatic.

[LauricAcid]

I don't understand why they are called "axioms"

I'm not expert in etymology, but Merriam-Webster traces the word to Greek for 'honor' and 'worthy'.

the term "postulate" is less likely to result in confusion"

That might be so in the wider context outside of mathematics. But once you study just a little bit of mathematical logic, there is no need for confusion as to the use of the term 'axiom' for formal systems. Anyway, mathematics does not even need the word 'axiom' or 'postulate'. One could simply refer to a set of well formed formulas and its consequences and avoid any parlance at all.

since the Peano postulates aren't self-evident truths

I won't argue that they are, but it should not be lost that, many people feel that at least with respect to the standard model, the Peano axioms are about as self-evident as anything could be said about natural numbers.

they are not part of an overall integrated epistemology

Mathematics is not philosophy, yes, that's true. On the other hand, if one wants to derive theorems of mathematics from Objectivist axioms, then there's nothing stopping one from attempting to do so.

but are based on the false assumption that you can extract certain gobbets of knowledge and treat them in isolation from other gobbets of knowledge, and when you discover that you really cannot construct these well-defined boundaries that result in "pure mathematics" with no reference of real-world entities, then you have to introduce pseudo-axiomatic statements so as to avoid facing the connections between this aspect of knowledge and that aspect of knowledge. Still, that doesn't make such postulates axiomatic.

That's just a lot of rhetoric, especially the idea of "constructing well-defined boundaries that result in "pure mathematics" with no reference to real-world entities" is a strawman unless you can show that the mathematics you object to depends on constructing well-defined boundaries that result in "pure mathematics" with no reference to real-world entities.

That [-1 is not in the theory of Peano arithmetic anyway, and even in the larger theory of set theory, -1 precedes 0 in the standard ordering of the integers, but 0 is not result of the set theoretic successor operation applied to -1] is certainly not self-evident"

It's evident if you can read mathematical formulas and follow a thread of theorems and definitions.

in fact is seems downright arbitrary."

Only to the extent that the axioms of ZF as a metatheory are arbitrary. If anyone can propose a "non-arbitrary" axiom set, then they're welcome to do it.

It is obvious that 0 is the successor of -1 just as 3 is the successor of 2."

No, as I said, 0 follows -1 in the standard ordering of the integers, but the successor operation applied to -1 is not 0. The successor operation is a specific set theoretic operation and is not what you think it has to be to conform to the English language sense of what a successor is. And again, if you don't like mathematics using words like 'successor' in a non-everyday sense, then fine, the word 'successor' itself is not needed in mathematics as it is just a handle for a formal mathematical operation (primitive in Peano arithmetic and defined in set theory).

I assume then that there is just this arbitrary stipulation for come kinds of mathematician that the term "successsor" is undefined for the predecessor of zero.

Peano arithmetic is just an account of the numbers 0, 1, 2, ..., so, of course, the entire theory will be whack to you if you don't accept that. But mathematicians don't proclaim that Peano arithmetic is the entire theory of mathematics. On the contrary, Peano arithmetic is taken in a wider context as it is one of many theories (such as the theory of the compelete ordered field (the real numbers)) to be compared among themselves in the larger theory that is set theory. So, since Peano arithmetic is just to account for the natural numbers, it is an axiom that 0 is not the successor of any natural number.

I'm not saying you can't make these kinds of arbitrary claims, just that it is not self-evident and therefore not axiomatic.

Fine. Mathematical logicians are the first to recognize that the Peano axioms are not true in all models. The use of 'axiom' in the context of formal theories does not pretend to carry a conviction that the axioms are self-evident (though, for certain theories, one may in addition to the mathematics, opine, philosophically, that certain axioms are self-evident). If the word 'axiom' bothers you, mathematics would not be changed a bit in its meaning if 'axiom' were replaced with 'schmaxion' or 'schpostulate' or whatever sound or combination of letters from the English alphabet aren't taken by some other term in use in mathematical parlance. And again, this is just parlance. Edited September 27, 2005 by LauricAcid

[LauricAcid]

CORRECTION: I wrote, 'It turns out that there are models of the Peano axioms that have objects that are not natural numbers, but this is a result of model theory, not something that can be stated within Peano theory itself.'

Please consider that to be striked out.

[jrs]

Indeed, -1 has the successor 0, so either the Peano Axioms are false, or "natural number" is a non-self evident term of art.

Suppose that I have a basket which is empty. It contains zero pebbles.

Suppose that I put a pebble into the basket. Now it contains one pebble. So one is the successor of zero.

Suppose that I put a pebble into the basket. Now it contains two pebbles. So two is the successor of one. Et cetera.

What is this "-1" of which you speak? What condition of the basket would allow me to place a pebble into the basket; and then have it be empty?

[DavidOdden]

What is this "-1" of which you speak? What condition of the basket would allow me to place a pebble into the basket; and then have it be empty?

So are you saying that the concept of "natural number" is self-evidently defined as "an operation analogous to putting pebbles into a basket"? And not, for example, the operation of adding or subtracting money from an account (presuming you're heard of the concept "debt")? I reject your definition of natural number as being self-evidently correct, although if you want to stipulatively define the term that way, or simple define another expression such as "positive integer", you can. It just isn't self-evident.

[LauricAcid]

What is claimed to be self-evident (whether this claim is justified is another question) is not that the sound 'NAA-CHUR-UL NUM-BUR' or the letters 'natural number' must refer to the counting numbers starting with zero, but rather the axioms as they pertain to the counting numbers starting with zero.

Edited September 27, 2005 by LauricAcid

[robstercraws]

The Peano axiom that I remember is "Zero is a number". That statement most certainly presupposes a number of more fundamental ideas, among them existence, identity and consciousness.

I'm not sure how you could logically support that statement considering you can recreate all of mathematics without every mentioning existence, identity or consciousness. Any textbook on the subject such as Bertrand Russell's Introduction to Mathematic Philosphy will easily show that.

Also, yes you can verify the validity of an axiom but not when it is the assumed proposition of the argument. Doing so is the very definition of circular reasoning.

For instance, you could prove certain events in the Bible to be valid such as the birth of Jesus through all sorts of scientific and historic evidence. But what you can't do is to use the Bible as part of its own validation process otherwise you get circular reasoning. So the moment you accept these Objectivist statements as axioms and therefore true by defintion, any attempt to use them in your validation process automatically produces circular reasoning.

Also, LauricAcid is absolutely right when she said that words have different meanings in different arenas. That is exactly my point. I was simply pointing out that the word "axiom" used in this discussion is different that what is meant in formal logic. Many people see the word axiom and believe that it has the same meaning, carries the same formal weight and often use it in arguments as if they did have these meanings. As long as you know that in this useage it means only to be assumed propositions and nothing more, then that's fine.

[DavidOdden]

I'm not sure how you could logically support that statement considering you can recreate all of mathematics without every mentioning existence, identity or consciousness.

The fact that you can easily ignore foundational principles dosn't mean that the idea doesn't depend on them. Existence, identity, consciousness and vast numbers of other concepts are taken for granted in mathematics, precisely because that are truly axiomatic and thus only need to be mentioned if you intend to do a complete logical reduction of mathematics.

[dougclayton]

So the moment you accept these Objectivist statements as axioms and therefore true by defintion, any attempt to use them in your validation process automatically produces circular reasoning.

As long as you know that in this useage it means only to be assumed propositions and nothing more, then that's fine.

This is a misunderstanding of the Objectivist axioms. They are most certainly not "assumed propositions for purposes of argument" (as you said in an earlier post). They are not a convention; they are irrefutable. Try it--you'll have to grant their validity to make any meaningful assertion.

[BurgessLau]

I'm not sure how you could logically support that statement considering you can recreate all of mathematics without every mentioning existence, identity or consciousness. Any textbook on the subject such as Bertrand Russell's Introduction to Mathematic Philosphy will easily show that.

What do you mean by "logically" here?

Also, LauricAcid is absolutely right when she said that words have different meanings in different arenas. That is exactly my point. I was simply pointing out that the word "axiom" used in this discussion is different that what is meant in formal logic.

Except for proper nouns, why do you hold that words have meanings? Is there some element of Objectivist epistemology that you are basing your belief on? If so, what is it?

[robstercraws]

This is a misunderstanding of the Objectivist axioms. They are most certainly not "assumed propositions for purposes of argument" (as you said in an earlier post). They are not a convention; they are irrefutable. Try it--you'll have to grant their validity to make any meaningful assertion.

That is exactly my point, the burden of proof rests on proving an assertion without using your beginning propositions not disproving them. Your statement is identical to saying that Leprechans exist because I can not disprove it wiithout using the concept of Leprechan. Science doesn't work that way, you must carry the burden of proof.

Also, I'm interested in your reasoning of why you say these are irrefutable concepts? Is it because we assume them to be true based on the concept of axiom or because of your belief in their validity? You can't have it both ways simultaneously.

BurgessLau does make a good point about words as meaningful. It's a very interesting and worthwhile discussion but one that perhaps could be addressed in as a different forum. And lastly by "logically" I mean only that the concepts mentioned play no role in the formal development of abstract mathematic concepts. Perhaps, she was talking about instances of these abstract concepts which is quite different and could indeed involve these concepts. I must add that I found your questions very insightful. Thanks for the input.

[AisA]

That is exactly my point, the burden of proof rests on proving an assertion without using your beginning propositions not disproving them. Your statement is identical to saying that Leprechans exist because I can not disprove it wiithout using the concept of Leprechan. Science doesn't work that way, you must carry the burden of proof.

But what is proof? It is facts, data, evidence, reasoning etc -- all of which presupposes existence (if nothing exists, there would be no facts), identity (if things have no identity, then we could not gather any evidence about them, since they would not be anything in particular) and consciousness (if there is no consciousness, there is no faculty for gathering or judging any facts, evidence, etc).

Any attempt at proof, any claim to knowledge, makes use of and is based on these three axiomatic concepts: existence, identity and consciousness.