“Axiomatic systems are riddled with a number of insurmountable difficulties.
Not least are the consequences of the discovery in 1933 of Godel's theorem.
Godel's theorem is demonstrable as a mathematical proof and hence can be
Without more substance to inform the thrust of this criticism than what
lordbyron provides here, it is simply not possible to debate it. Then again,
what were Godel’s own premises? I have studied some of Godel's work, but I
admit I was not very impressed by his philosophy. Dr. Harry Binswanger, an
Objectivist philosopher, in his elaborate lectures on the nature of
consciousness, offers some very interesting points in regard to Godel’s
ideas. In his discussion of those ideas Binswanger exposes Godel’s own
reliance on stolen concepts and the fallacy of pure self-reference. So I
would say right off that Godel, whether one wants to call him a genius or
not, could very well have been mistaken in some of his views. It is possible
for one to be a genius, but also misguided.
The need for axioms is implicit in all knowledge, for knowledge is
hierarchical in nature. Whenever someone states that "X is true *because* Y"
he is implicitly acknowledging this hierarchical nature of knowledge. When
one asks you why you believe something you say you believe, he is implicitly
acknowledging that knowledge has hierarchical structure. The same is the
case when someone asks "what basis do you have for concluding this?" Etc. An
axiom is a conceptual starting point: it serves to ground a philosophical
system. As such, an ultimate axiom must be irreducible to prior statements,
and axioms themselves, if they are propositional in form, are reducible to
constituent concepts which themselves cannot be reducible to prior concepts
(for then they could not serve as axioms).
”Basically any axiomatic system is unable to demonstrate the truth of
everything which is contained within the system.”
This is a little vague, so I’m not certain exactly what it is trying to say.
I do not think that the task of axioms is to “demonstrate the truth of
everything which is contained within the system” as such, but to provide an
objective foundation to the system so that one can identify truths and
validated knowledge on a consistent basis.
“The consequence of this is that you could have a logically rigorous and
watertight argument with sound axioms and valid reasoning and yet there
would inevitably and necessarily be parts of the universe which you would be
unable to discuss.”
Well, this kind of charge can always be launched against any system which
does not claim omniscience on the part of knowers, since when one is not
omniscient, there is always going to be something in the universe which he
does not know. But how this constitutes a legitimate point of criticism
against an objective approach to philosophy is not stated here.
“Even if you add more axioms to broaden the base of your system you could
never have sufficient initial premises to cover all that the universe
contains. This is known as Godel's incompleteness theorem.”
I think the applicability or validity of this claim depends on the nature of
the axioms in question and the breadth of their reference. I do not see how
this point can serve as a durable criticism of Objectivism, since the
fundamental axiomatic concept in Objectivism is the concept 'existence',
which is the widest of all concepts (it applies to everything which exists),
and is therefore literally universal (since 'universe' is the sum total of
that which exists).
”To discuss the universe in its entirety with an axiomatic system requires
inconsistency. Yes thats right, an axiom that is inconsistent with itself or
another axiom within the same system. Known as Godel's inconsistency
If this is what Godel thought, then I’d say he was mistaken. In fact,
Objectivism was not around when Godel was writing, so he wrote about axioms
in ignorance of Objectivism. Furthermore, lordbyron does not inform us why
an axiomatic system must be inconsistent with itself in order to discuss the
”Of course, it is not just the process of argument which suffers in an
How exactly does “the process of argument… suffer in an axiomatic system”?
The very process of argument implies the need for a starting point.
Otherwise, the system from which one is arguing is at the mercy of an
infinite regress. I see no good reason why one should allow himself to fall
into such unnecessary traps.
“There is also the difficulty of presenting a universally accepted axiom.”
An axiom does not gain validity simply because it is universally accepted.
So I don't know why the criterion of universal acceptance is even relevant.
lordbyron does not say.
“Axioms have to be internally consistent.This is not difficult to achieve.”
Internal consistency is not the problem; rather, is the axiom (or, more
properly, the axiomatic concept) in question truly irreducible, and on what
is it based? These are questions which systems which propose axiomatic bases
should be prepared to answer.
“Any axiom which purports to say something about the universe has to show
why the existential content can be assumed to follow from the theoretical.
Not at all easy.”
But what if the system in question does not assume that the “existential
content can be assumed to follow from the theoretical”? If existence exists
independent of consciousness, as Objectivism holds, then it is incoherent to
asume that "existential content" should "follow from the theoretical." I
think this is a fundamental reversal.
”For those who want to explore the existential content of axioms further,
take a look at Euclid. His axioms stood for 2000 years and were heralded as
perfect knowledge, a level of truth to which all knowledge should aspire.
They were considered to apply to the universe as they dealt with
3-dimensional space. Internally at least, the axioms are consistent.
Applying them to the universe became problematic only when Einstein decided
space might be curved. Perfect knowledge was shown to be far from perfect.
Indeed we now know it is possibly wrong.”
Simply because some point which Euclid proposed as an axiom might, after
centuries of thinking otherwise, be determined to be mistaken, does not
necessarily mean that the axioms of another system fall prey to the same
errors, or that the concept of axioms as such is invalid.
Besides, as I understand it, Euclid's system is still valid for plane
geometry (i.e., geometry on surfaces which are not curved). Non-Euclidean
geometry's validity for measurement of forms on curved surfaces does not
invalidate a system which is valid for measurement of forms on surfaces
which are not curved. The two systems are not contradictory to each other
because the context and purpose of their application differ from one
”So Steel, I make no claims to be a genius but there are many who would
class Godel as a genius. He was able to show a revolutionary new theory (at
least it was new in 1933) demonstrating the inadequacy of axiomatic
Specifically, which axiomatic systems did Godel review?
”Anybody who wishes to claim otherwise (such as yourself) either
demonstrates that they are writing before 1933 or that they have not read
and understood one of the most important mathematical discoveries of the
These alternatives are not exhaustive. For indeed, it may be possible that
someone has examined Godel’s criticisms and has found them to be
inadequately developed or insufficiently reasoned (as Dr. Binswanger has
“As I assume you are writing in the 21st century I can only conclude that
you are ignorant of the facts and are way out of your depth.”
I wager that since Godel was writing in the early 20th century he was
ignorant of Objectivism.
“Its easy to believe anything if you pick and chose the facts to fit the
Yes, I think that’s true.
”I would like to see if anybody has an axiom they consider to be irrefutable
and applicable to the universe.”
Objectivism: existence exists. Since the universe is defined as ‘the sum
total of existence’ this axiom logically applies to the totality, since the
totality in question is existence as such. Just by accepting the fact that
the universe exists, one implicitly affirms the Objectivist axiom.
Furthermore, even to attempt to refute the Objectivist axiom, one must
acknowledge that this axiom exists, and thus affirm its content as well. It
is irrefutable, and universally applicable.
lordbyron offers the following 7 axioms from Spinoza. I would say that none
of these statements can qualify as actual axioms, since each one is made up
of numerous concepts, and thus these statements cannot be said to be
irreducible. A legitimate axiom itself is a statement which is reducible to
axiomatic concepts (e.g., the statement "existence exists" is reducible to
the axiomatic concept ‘existence’), and these concepts (e.g., 'existence')
cannot be reduced to prior concepts. That’s why I simply grin when the
“Scripturalists” (a la Gordon H. Clark) claim that their axiom is “The bible
alone is the word of god” since this “axiom” consists of numerous concepts,
none of which are irreducible or perceptually self-evident. See Ayn Rand,
“Axiomatic Concepts,” in her book _Introduction to Objectivist Epistemology_
for more details in this regard.
lordbyron quotes B. Spinoza’s axioms as follows:
“I. Everything which exists, exists either in itself or in something else.”
This seems incoherent to me. What does it mean for something to “exist in
itself”? Either something exists, or it does not. I can understand “exist as
itself,” but how does something “exist in itself”? A thing is itself. And
why the dichotomy, “either it exist in itself, or it exists in something
else”? If something can exist "in itself" (whatever that means), can it not
also exist in something else as well? Why are these two conditions assumed
to be mutually exclusive?
“II. That which cannot be conceived through anything else must be conceived
What exactly is this saying, and why should one accept this as true?
”III. From a given definite cause an effect necessarily follows; and, on the
other hand, if no definite cause be granted, it is impossible that an effect
Again, this cannot be axiomatic, for it consists of numerous concepts, and
thus as such it is not irreducible. Also, it seems to assume a Humean rather
than Aristotelian conception of causality, and this preference needs to be
argued for, I would think.
”IV. The knowledge of an effect depends on and involves the knowledge of a
Hmm…. I know that my microwave can heat up my tuna melt (the effect), but I
do not have a lot of knowledge of how it works (the cause), so I think this
statement is questionable. Indeed, in many cases, we must begin with our
knowledge of an effect in order to infer the nature of its cause. Perhaps
I’m mistaken here?
”V. Things which have nothing in common cannot be understood, the one by
means of the other; the conception of one does not involve the conception of
A statement like this simply causes me to ask, “why?” which in my assessment
is sufficient to dispute its nature as an axiom. It’s just not self-evident.
And what in existence has virtually nothing in common? At the most
fundamental level, all things which exist have something in common, and that
something is the fact that they exist.
”VI. A true idea must correspond with its ideate or object.”
I understand this statement to be one which attempts to isolate the
necessity of a legitimate concept’s reference either directly to reality
(such as in the case of the axioms) or to other concepts which ultimately
find their basis in the facts which we perceive. But even if we agree here,
this statement is not axiomatic in nature, for it is not irreducible, nor is
it perceptually self-evident.
”VII. If a thing can be conceived as non-existing, its essence does not
This statement relies on the existence-essence dichotomy, which I reject.
Existence is metaphysical, while essence is epistemological. In Objectivism,
this distinction (as I correct it here) is not a dichotomy, since
Objectivism does not hold that existents have an essence in
contradistinction to its existence or identity (as Aquinas thought; indeed,
he needed this dichotomy in order to argue for the reality of miracles).
Rather, essence is an abstraction which isolates the primary characteristics
of objects for the purpose of concept-formation. Furthermore, what one can
conceive or not conceive is irrelevant to the facts of reality (cf.
“ontological argument”). I can conceive that the moon is made of green
cheese. But reality does not conform to this simply because I can conceive
of this. In this point and in many others, Spinoza assumed the primacy of
consciousness view of reality, which invalidates his ideas.
”Question 1: What is a 'stolen concept'? I've never met this term before and
wonder if it is a label you have formulated or whether it is an Americanism.
Can't find any reference, so would appreciate a definition/explanation”
That’s fine. The fallacy of the stolen concept occurs when one asserts a
concept while denying or ignoring its conceptual roots, or the facts which
make the concept possible in the first place. For instance, if I were to say
that I am an expert in algebra, but that I deny the validity of 2+2=4, I
would be “stealing” the concept ‘algebra’ from its proper place in a
rational conceptual hierarchy, since algebra is not possible without the
validity of its arithmetic basis. The error which the stolen concept fallacy
commits is the disregard for the proper hierarchical relationship a concept
has to those concepts on which it depends. A concept is asserted, but a
concept or concepts on which it depends have been denied. This is
The following links will connect you to articles which discuss the nature of
this frequently overlooked fallacy:
“Question 2: Can you do anything CV without using your mind? Scholastic
babble is meaningless and drivel however you attempt to dress it. If you
were so confident in your beliefs, (or lack of them if you prefer) you
wouldn't descend to this form of nonsense.”
No, I cannot do anything without using my mind. I agree that “scholastic
babble is meaningless and drivel however [one] attempt[s] to dress it.”
Indeed, I am no scholastic, and what I write is not “babble” simply because
lordbyron does not understand what he reads.
As for the statement, “If you were so confident in your beliefs, (or lack of
them if you prefer) you wouldn’t descend to this form of nonsense,” I’m not
certain what you’re referring to. Where do I “descend” to a “form of
nonsense,” and what exactly have I written that you consider nonsense, and
why? Again, if you do not understand it yourself, that alone is not
sufficient to term something nonsense, is it?