The Meaning of a Sentence, Meaninglessness & Reductive Regress

 

I think that there’s a problem with your use of the word “meaningless”, as in


“’God exists’…is meaningless."


“God exists” has a meaning; or, at least, it makes sense. You may of course be using the word “meaningless” in a specific and precise technical sense. (Though there are very many rival philosophical conceptual schemes, as you know.)


For example, “God exists” may be meaningless because it can’t be verified (in the language of certain early logical positivists) , or it makes no difference to experience (in the language of certain pragmatists), or it has no observational consequences (in the language of Carnap). In that case, it may be better to say that the expression “God exists” is, at best, philosophically problematic, or, at worst, logically incorrect. Again, I don’t think that it is meaningless.


Some philosophers, for instance, say that


“God exists.”


Is equivalent to or means


“Godhood is instantiated.”


But “God exists” doesn’t seem to mean that at all. It may be the case that the statement “God exists” should be “Godhood is instantiated” if it’s to be philosophically coherent and acceptable. But “God exists” simply doesn’t mean “Godhood is instantiated”. The statement “God exists” means God exists, in a manner of speaking. Similarly, perhaps “Godhood is instantiated” isn’t even the correct “logical form” of the “surface grammar ” either 


The distinction to be made is that when a philosopher says that “Godhood is instantiated” (or another version) is equivalent of or means God exists in correct logical grammar, or that it is equivalent to it, he is doing, alternatively, normative, or stipulative, or revisionary (or even “transcendental”, according to Stephen Yablo) metaphysics. That is, the philosopher who opts for “Godhood is instantiated” rather than “God exits” isn’t doing “descriptive metaphysics”.


These two terms, “descriptive” and “revisionary metaphysics”, are taken from Strawson (though the idea isn’t exclusively his). This is what Strawson himself wrote in his well-known book Individuals:


“Descriptive metaphysics is content to describe the actual content of our thought about the world, revisionary metaphysics is concerned to produce a better structure.” (page 10)


For example, take your reference to “red exists” (seemingly a reference to a universal). We can have


“Red is a colour.”


parsed (or analysed) into the nominalist


“Red things are coloured.”


according to one revisionary metaphysical scheme (to be argued for). Or we can have


“Red is a colour.”


parsed (or analysed) as


“Reds are colour tropes.”


according to another revisionary metaphysical scheme (again, to be argued for).


Though the statement “Red is a colour” doesn’t mean Reds are colour tropes and it isn’t equivalent to “Red things are coloured” either. You could argue for the “Godhood is instantiated” version of “God exists”; though I don’t think that the latter is meaningless.


So I assume that you mentioned “red exists” because it is seemingly a reference to a universal, and is therefore meaningless because universals are deemed to be non-spatiotemporal entities.


The strange thing is that many philosophers have had a problem with the statement “God exists” not because of the proper name “God” (e.g., not referring, etc.); but because the predicate “exists” is, as it were, non-referring or vacuous. That is, the predicate “exists” adds nothing to the subject term “God”.


For example, it's argued that


“Tigers exist.”


has the same grammatical form as


"Tigers are striped.”


though it has a different logical form. That is, according to the revisionary metaphysician (even if he doesn’t accept that description), “Tigers exist” is logically incorrect. That may be the case; though “Tigers exist” is still meaningful – it still makes sense.


Just to set the cat among the pigeons, D. F. Pears not only argued that the statement “God exists” makes sense, that is grammatical sense: it also makes logical and philosophical sense:


“[To say that] the verb ‘exist’ does not add anything to the concept of the subject [e.g., ‘God’ or ‘red’]…is false: for to say that a concept has instances [or is ‘instantiated’] in reality is certainly to add something to it…” (from ‘Is Existence a Predicate?’)


Just to show how reductive the belief in a correct logical form can go, let’s get back to


“God exists.”


this can be said to mean or be equivalent to


“Godhood is instantiated.”


which in turn can be said to mean or be equivalent to


“There is at least one x, such that x instantiates Godhood."


And this can be said to mean or be equivalent to


x (Gx)


So now we have this strange identity statement:


“God exists” = x (Gx)


It could now be said (though I have a feeling it may be wrong) that “God exists” and "x (Gx)" have different Fregean “senses”; though the same “reference”. That is, someone may believe the statement “God exists” is true (or false)though not believe the same of "x (Gx)".


*) Just as some philosophers argue that we can offer the correct logical form of certain contentious - though grammatical - statements; so too could certain grammarians argue that they can offer the correct grammatical form of:


Buffalo Bill’s
defunct
who used to
ride a watersmooth-silver
stallion
and break onetwothreefourfive pigeonsjustlikethat


Jesus
he was a handsome man
and what I want to know is


how do you like your blueeyed boy
Mister Death


(e.e.. cummings, ‘Buffalo Bill’s’)


In this instance, the analysis, interpretation, parsing, etc. of the above may be acceptable. Certain philosophers think that we are capable of giving exact literal equivalents of metaphors and other poeticisms. However, would the poem above mean the same as its interpretation? Would the interpretation be equivalent to the poem? (What would we mean by “equivalent”?) 

Brentano on Mental Acts & Mental Objects

Franz Brentano offers us a view of mental representation that is an example of self-consciousness.


We don't just ‘have a representation’, or ‘see one’, or ‘become aware’ of one, we are also at the same time aware of the representing. Prima facie, it seems strange that Brentano argued that we ‘represented’ a representation to ourselves, rather than simply had one. The way he puts the representing of representations is that it's a cognitive or volitional process. It's often thought to be the case, say, with sense impressions and sense-data that they are just ‘given’ to us without cognitive activity. That is why the term ‘the given’ was so important to 20th century empiricist epistemology. That very term stressed the supposed fact that sense-data aren't the result of any cognitive activity and therefore must be unpolluted by our prior concepts or categories.
 
For instance, when I look at my white fridge, seemingly non-cognitively, white, square (not literally square!), and other kinds of sense-data are just given to me – I simply have them or, in a way, find them in my mind [see Ned Block’s 1992/1997]. All this seems to distinguish sense-data from Brentano’s ‘representation’ because he argued that we cognitively ‘represent’ the representation to ourselves.


Of course we must first ask what it actually means to ‘represent a representation’. Can a representation also represent something or other without our own cognitive representing of the representation that itself represents something external? ‘Representing’ implies that we make some kind of cognitive decision to become aware of a representation that would otherwise be un-cognised (even if an un-cognised representation makes no sense!).


Take the representation of a particular sound. When we have a sound-representation, we don’t just have it, or become aware or conscious of it, there must be a secondary cognitive process that becomes part of the sound-representation and all conscious representations. We must also be conscious of ‘the act of hearing it’. In that case, the sound-representation must come along with what must be another representation or awareness – that of the act of awareness itself. So we now have:


i) the sound-representation


ii) the act-representation (of being conscious of the sound-representation).


In that case, all representations must have binary components. We aren't only conscious of the representation itself: we must also be conscious of the act of representing that representation to ourselves. We can say that the act-representing is of a higher order. And the representation-object of the act is of a lower order. Though, in all genuine representations, that higher-order act must always accompany each conscious representation.


Brentano now stresses the fact that although representations are of this binary nature, each conscious representation doesn't also require two acts. The representation itself is seen as the ‘first object’ by Brentano. And the mental act of representing, or being conscious of representing, is the ‘second object’. The second object, or mental act, can be deemed as ‘reflexive’ in that it's a mental process or act which has its own object, which itself becomes the object of a reflexive act of awareness about the mental act itself. In that case, couldn’t we also become aware not only of the mental act with a representation as its object, but an act which has another mental act, rather than a representation, as its object? This third-level process would include two acts of cognition, where one mental act itself becomes the mental object of a third-level mental act. This process, clearly, could continue indefinitely. However, despite that possible higher level indefinite regress, Brentano himself is aware of the possible lower-level ‘Cartesian presumption’ that would lead to an endless regress of mental acts. This, however, is about the lower-level relation between a conscious act of awareness and a representation not to be deemed a mental act in itself; but only a mental object of various cognitive acts. That is, if the initial representation were itself an act, though still a mental object of a higher mental act, then that representation-act, or mental object-act, would itself have its own mental object or representation. We would have to move one step further beyond what we took to be the pure representation and non-cognitive starting point to its own representation. But if we go further than the initial pure representation, then we could also see the initial representation’s own representation-object as also being a representation-act as well. This process, clearly, could go on indefinitely, just as my higher-level version could.


So the ‘Cartesian presumption’ must be that we must reach the foundation or bedrock of such mental processes at some time. Therefore some representations, or Cartesian ‘ideas’, must be pure representations and not at all representation-acts.


This train of thought is analogous to Wittgenstein’s position in his Tractatus that there must be ‘logically proper names’ that are no longer ‘analysable’ and must therefore be basic and fundamental. (These names too must be seen to halt a similar indefinite regress, just as his ‘simple objects’, the objects of the ‘simple names’, called a halt to a possible ontological and logical indefinite regress.)


However, if we call a halt to the Cartesian indefinite regress by accepting the existence of a pure representation or representation-object, then what of our higher-level case of an indefinite regress of mental acts, or higher-order acts which take lower-order acts as mental objects; though which are in fact still act-objects and never pure representation-objects. In the higher-order case we can have an act of an act of an act… And so on. Just as in the Cartesian case we could have a mental act of a mental act… All this is instead of an act which has as its object a pure representation that is in no way itself a mental act. If we can call a halt by positing a lower-order representation, perhaps in our higher-order version we could take a specific higher-order mental act as the end of a short regress, as it were. The lower-order representation would then therefore be foundational and fundamental, whereas our higher-order mental act would be the end-point or final consequence of a small foundational structure supported by our pure representation.


In the case of the earlier sound-representation, we can take our higher-order approach of taking a possible indefinite regress of mental acts rather than the lower-order Cartesian approach of the indefinite regress of trying to find a representation that isn't also a mental act which has its own object-representation. In terms of being aware of an act rather then of a representation, it would mean that


to be aware of a sound is to be aware of being aware of a sound, and then to be aware of being aware of a sound would involve being aware of that awareness, and so on indefinitely. We can now ask: How do we, or how can we, call a halt to this potential indefinite regress? Would we need to treat a higher-level act in roughly the same way as we treated our foundational representation? Instead of seeing a higher-order mental act as foundational, we could see it as an end-point or some kind of cognitive brick wall which we couldn’t break through.


Brentano himself calls a halt to this potential regress of acts in this way.


Firstly, he rejects the higher-order act of being aware of our awareness of a sound. This means, quite simply, that awareness itself can't be a mental object of another mental act of awareness. In other words, all we really have is a second-order acts and first-order, or foundational, representations. We must stick with this binary relation and not believe that there are also tertiary or yet higher-order cognitive acts which take other cognitive acts as their mental objects. However, at a prima facie level, it does seem possible to be aware of an instance of our awareness, or to take a cognitive act as an object-act, rather than as a pure object or representation.


How does Brentano defend his sudden halt to higher-order acts of awareness?


Brentano argues that this ostensible case of an act of awareness which takes another act of awareness as its cognitive object is in fact a different kind of a mental act from the lower-order act of simply being aware of a sound. So, according to Brentano, the ostensible act of being aware of being aware of a sound is no different to the more basic being aware of a sound. At a prima facie level, again, the third-order awareness of an act of awareness seems possible. How does Brentano actually defend his rejection of third-order acts of awareness?


If we were to ‘observe a mental act’ it would be an attempt to make it an example of Brentano’s ‘first object’. In that case, we would find ourselves with two first objects – the initial pure representation and an act of awareness of that representation which is now itself a ‘first object’ of another mental act of awareness. Clearly, we can't have two first objects. Brentano therefore concludes that an act, rather than a pure representation, could never be taken as a first object but must always be taken as a ‘second object’.


Even though we are talking here about mental acts and mental objects of those acts, Brentano rather strangely brings in the notion of ‘observation’. He makes the point that when we talk of observation we presume a distinction between observer and observed. Clearly, in that case of a pure representation and its act, the representation is analogous to the observed and the act of awareness of that representation is analogous to the observer. What would be an act of awareness of another object? We would therefore have two acts or processes of awareness occurring at the same time, each with its own mental object. If we think in terms of two cognitions or two sub-vocal expressions of two different propositional beliefs occurring at the same time, such scenarios seem even more unlikely. Take two examples:


i) Tony Blair is a liar.


ii) Venus is a planet.


Such sub-vocal expressions could not occur at the same time in the same subject. This becomes even clearer if we say that the various objects, i.e., Tony Blair, Venus, planets, liars, etc. must somehow be cognised at more or less the same time. Not only that: two different mental acts of predication must occur at the same time. We couldn't observe a mental act that is just as much an act or process as the mental observation itself. In this case, an act of observation and an act of awareness must occur at the same time. So an act can't be a first object because it is not in fact a genuine mental object at all – it is a mental act or process. An act or process cannot be the mental object of a further mental act because one act would need to occur at the same time as another mental act.


So we were wrong to see mental acts as becoming the possible mental objects of further mental acts. We assumed that an act or process could itself become the mental object of another mental act. We failed to realise that an act, even as seen as a mental object, must still be a mental act in the scenario of an act of awareness that takes another act of awareness as its mental object or representation. According to the first case, the representation was the observed and the act was the observer. But now we have two acts, even when one act is taken to be a mental object-act. If we have two acts, then we must have two observers and no genuine example of a pure observed representation. Prima facie, the idea of an observer observing an observer seems odd. However, it does actually seem possible, as in the case of a spy of one country spying on a spy of another country. In that scenario, why can’t a mental act take another mental act as its mental object, or even as its quasi-object? Again, an observer can take another observer as his object of observation. So just as an observer can take on two roles, as observer and observed, why can’t a mental act also be a mental object for a further higher-order mental act? However, two cognitive mental acts cannot occur simultaneously. In that case, it's simply incorrect to see a mental act as also a potential mental object of another mental act. Unlike a representation, a mental act is a kind of mental process. A mental process can take a representation as its mental object because the representation is not itself a process. However, in order for a mental act also to become a mental object-act, we would need to accept two simultaneous and different mental processes occurring at the same time in the same mind. Clearly this couldn’t happen. For example, we would need to become aware of a mental act which must also, as a mental process, itself be in the mind if taken as the object-act of a further mental act. It should have been clear that this higher-order mental process would require a mental act, better seen as a temporal mental process, occurring at the same time as a mental act or temporal process in the mind which takes it as its mental object. But that higher-order mental act couldn't take an act as a mental object if that act itself is a temporal process of awareness of a lower-order mental object. And if this higher-order mental act can't actually be a genuine first object, then the yet higher-order mental act we believed took such a lower-order mental act as its mental object couldn't itself be a genuine higher-order mental act simply because it doesn't actually have its own higher-order mental object-act. Such ostensible higher-order mental processes and mental objects would therefore be like a single person playing the piano at the same time as doing a crossword or drinking a cup of tea and a can of lager concurrently. Looking for such higher-order mental objects and mental acts is also like a dog trying to catch its own tail or trying to look at our own eyes without the benefit of a mirror or reflector.


Conclusion: Brentano and Introspection


Finally, there may be a problem with Brentano’s use of the word ‘observation’ to get his arguments across. It depends, however, if he really believed that mental acts effectively - or literally - observe representations, or whether it's just an analogy on his part. Clearly, ‘observation’ is a term connected to the faculty of vision, which couldn't be applicable to any mental process or mental act. If Ryle, amongst others, rejected ‘the eye of the mind’ [Ryle, 1949], or the ‘internal theatre’ [Dennet, 1991], or introspective acts of perception of the mental events and mental objects on the mind’s stage [Davidson, 1980], so we should be equally suspicious of Brentano’s use of the term ‘observation’.


If it's not the case that mental acts observe mental representations, then how, in fact, can we make sense of the relation between an act of awareness and its representation-object, or of any other such introspective processes in the mind for that matter [see also Wittgenstein, 1953/57]? If we deny such quasi-visual metaphors, if they are metaphors in Brentano’s mind, then we may have to give up the notion of introspection itself, and even, as Ryle and Churchland do, the very ‘concept of mind’ as it is - or has been - seen traditionally.


Notes and Further Reading


Block, N – (1992/1997) ‘Begging the Question Against Phenomenal Consciousness’, in The Nature of Consciousness, ed. N. Block, A Bradford Book
Brentano, F – (1874) Psychology from an Empirical Standpoint
 - (1911) On the Classification of Psychical Phenomena
Davidson, D – (1980) ‘Knowing One’s Own Mind’, from his Essays on Actions and Events, Oxford University Press
Dennet, D – (1991) ‘The Cartesian Theatre’, in Consciousness Explained, Boston: Little, Brown, and Co.
 - (1981/1997) ‘Skinner Skinned’, in his Brainstorms, Penguin Books
Ryle, G – (1949) The Concept of Mind, Penguin Books
Wittgenstein, L – (1953/57) Philosophical Investigations

Theories aren't Always Intellectual Constructs

  
You seem to have a problem with the word "theory" (that is, Quine's usage of that term). Perhaps if you didn't take the term so intellectually or scientifically.

 
Take a term outside Quine's usage.


Virtually all philosophers of mind use the term "folk psychology". And virtually all these philosophers deem folk psychology to be "theoretical" or a "theory". Folk psychology is a theory upheld by the folk, or the Man on the Street; though that doesn't thereby mean that every member of the folk sat in his or her armchair and devised a theory of mind or even of his own psychology.


Here's Paul M. Churchland on the matter:


"Not only is folk psychology a theory, it is so obviously a theory…The structural features of folk psychology parallel perfectly those of mathematical physics; the only difference lies in the respective domain of abstract entities they exploit - numbers in the case of physics, and propositions in the case of psychology." (From "Eliminative Materialism and the Propositional Attitudes", 1981.)


The theory philosophers call "folk psychology" is more or less inherited. Indeed Churchland thinks that the theory goes back to the ancient Greeks and hasn't really changed much since then. Though it is still, despite its long lineage, a theory. However, it's not an intellectual construct as such; at least not in terms of each individual who adheres (non-cognitively) to it. What it is, despite that, is a scheme of interrelated and mutually supportive concepts, beliefs, truths, etc.


There are numerous such schemes or theories held by many of us without much theoretical, scientific or philosophical hard work on our own part. For example, an individual could accept a huge theory (say, an ideology, religion or philosophical system) without doing much cognitive hard work. The point is, however, that he accepts the theory or is simply born into it. A theory, theoretically, could include only, say, three interlinked and mutually supportive concepts and/or beliefs, etc. And even then they may be accepted as a package-deal and not be knitted together by each individual who accepts them.


Quine's use of the term "posit" (in your example) may put people off a little. But according to Christian theory in the Middle Ages, demons, angels, etc. were posited. According to scientific theory in a previous century phlogiston was posited. According to Aristotelian metaphysics the earth as being the centre of the universe was posited. And today "super strings", numerous particles of various description etc. may simply be posits.


As Quine said in one of your quotes, we can't ever be theory-less. We move around within conceptual schemes (large systems of inter-related theories). We can't achieve a view from Nowhere. Sometimes we may think or feel that we are theory-less (at least in certain respects) simply because we were born into particular theories or conceptual schemes. They now seem almost innate. But they are contingent. (However, certain ways of seeing the world may well be a priori. Say, for example, if one accepts a Kantian position on experience.)

Raymond L. Wilder on the Foundations of Mathematics

From the outside, the early 20th century obsession with the foundations of mathematics may seem strange. It may seem even stranger if we realise what the end result of this obsession was. According to Raymond L. Wilder, the modern mathematicians with


"his most powerful symbolic tools and his powers of abstraction and generalisation have failed the mathematicians in so far as 'explaining' what mathematics is, or in providing a secure 'foundation' and absolutely rigorous methods". (197)


It's quite remarkable that Wilder claims that the modern mathematician has failed to explain what mathematics is considering the fact that even the layman would have a good go at the job.


The question is: Why can’t they explain what mathematics is?


Why is this task so difficult?


Was it Gödel’s results that stopped mathematics from ‘providing a secure foundation’ as well as ‘absolutely rigorous methods’? Is it really the case that the search for foundations, as well as for absolutely rigorous methods, is well and truly over, let alone when Wilder wrote these words in 1968?


From what Wilder says next, it seems as if mathematics not having any foundations, or not being free from all contradictions, may not be such a bad thing. More precisely, he writes that
 
"perfect rigour and absolute freedom from contradictions in mathematics are no more to be expected than are final and exact explanations of natural or social phenomena". (197)


And, of course, in science we don't have "exact explanations of natural and social phenomena" and nor are such things ‘expected’ in science. Is this really the case in mathematics as well? Surely not. Perhaps this conclusion, on Wilder’s part, is simply a result of his materialist, sociological or even Marxist position on the practice and history of mathematics. Surely even these positions accept different standards from maths – indeed, they do.


Again, it is no surprise that Wilder says what he says if he that "the only reality mathematical concepts have is as cultural elements or artefacts" (197). This position seems to go even further than constructivism; though perhaps not as far as the late Wittgenstein.


More technically, Wilder expresses his constructivist, Marxist or sociological position on mathematics by elaborating on the notion of a ‘completed infinite’ (198). This sounds like a blatant and direct contradiction. How can any infinite be complete or completed? If it is completed, then surely it's not infinite. What, exactly, does Wilder say on this issue of the completed infinite? -


"For example, an infinite decimal is not something that ‘just goes on and on without end’. It is to be conceived as a completed infinite, just as one conceived of the totality of natural numbers as a completed infinity." (198)


Wilder gives us examples of completed infinities, the infinite decimal and ‘the totality of natural numbers’; though he doesn’t say what such things actually are or what the phrase ‘completed infinite’ means. The following hints at an explanation; though it doesn't help the non-mathematicians much. He writes:


"Symbolically, it may be considered a second-order symbolism, in that it is not susceptible to complete perception, but is only conceptually perceivable." (198)


Do you have a vague idea of what Wilder means by the above? Perhaps it's a kind of ‘direct insight’ or intuition into the nature of completed infinities. It can be conceptually perceived; though not seen – literally or even non-literally.

 

Gödel's Incompleteness Theorem & Leibniz’s Dream

Albert Einstein and Kurt Godel

The mathematician and educator, Morris Kline, once made a rather grand claim about Kurt Gödel’s Incompleteness Theorems when he (in his Mathematics: The Loss of Certainty) said that it

"was a response to Leibniz’s 250-year-old dream of finding a system of logic powerful enough to calculate questions of law, politics, and ethics".

Perhaps Leibniz’s dream had nothing to do with applying logic to the content of law, politics and ethics; but only to the form of the arguments in which these things were expressed. For example, in ethics, logic can't show us “what is good”. However, it can detect good and bad arguments as to what constitutes “the Good”.

Similarly logic can show faulty reasoning in political and legal debate; regardless of the actual content of these debates.

So, in that sense, it's indeed true that logic can be applied to law, politics and ethics – indeed to anything! So just as the premises of a deductive argument needn't be true in order for the argument to be valid; so the content of political, legal and ethical statements doesn't matter to the logician - though what follows from them, logically, does matter him.

Logic can "provide the tools to resolve ethical questions by mere calculation" if it dealt only with form and not with metaphysical, epistemological and semantic content.

In any case, were Gödel’s theorems really a response to Leibniz’s dream? Perhaps it was just Gödel’s way of showing us that, well, an axiomatic system (or mathematics generally) can't be both fully consistent and complete – that’s it (without philosophical knobs on).

Much has been made of Gödel’s theorem by non-mathematicians and by many non-philosophers. Morris Kline expresses much of this here. He writes that we

"might think that Gödel’s proof implies that the rational mind is limited in its ability to understand the universe".

How a result in metamathematics could do that (even in principle), I’m not sure. In any case, the mind, again in principle, must surely be limited in some way or ways. Perhaps that means that it could never understand everything there is to know about an infinite universe. Indeed this is bound to be the case. Only an omniscient mind could know everything there is to know about the universe.

Kline makes this point. He says that

"though the mind may have its limitations, Gödel’s result doesn’t prove that these limitations exist".

What is limited isn't the mind as such; but that "axiomatic systems are limited in how well they can be used to model other types of phenomena". This has nothing to do with the mind of man taken generically! It's to do with axiomatic systems and the modelling of other types of phenomena.

Not only that: the "mind may possess far greater capacities than an axiomatic system or a Turing machine". I would say that of course the mind does actually possess far greater capacities than an axiomatic system or a Turing machine. Evidently! For a start, the mind can create great poems or pieces of music. It has memory, experience, imagination, the ability to dream, create, invent, manipulate the environment and so on. Some of these things Turing machines can do; though many of them they can’t do. And no single axiomatic system or Turning machine can do all the things a human mind can do – not even a deranged or damaged human mind!

Another common supposed result of Gödel’s theorems is to assume that his proof implies a limit to artificial intelligence. Perhaps this is a more feasible idea because it must be about the mathematical limitations of artificial intelligence – and that would be relevant to Gödel’s proof. That is, would an indefinite advance in AI be halted by the result of Gödel’s proof which showed that if a mathematical system (therefore all combined) can't be both complete and fully consistent, then a project that relies on mathematics (that is, AI) will never be both complete and fully consistent? Thus there will be a limit to what AI can do.