I did not claim that there are no necessary truths or that the contingent Cogito can't grasp them. What I am claiming is that there is nothing "divine," or eternal about necessary truths. Necessary truths are simply tautologies, some simple, others more complex, like 5 = 4+1 or 5 = 20+10 - 25, etc. This is NOT the meaning of necessary, as I tried to explain it to you previously.

Truths would be necessary, or eternal, as per your definition of the term, only if they were thought by a necessary Cogito that I could experience in the first person, present tense mode, which I can't.

What I am claiming is that there is nothing "divine," or eternal about necessary truths. — charles ferraro

A distinction can still be made between necesssary and contingent, without referring to the "divine". Arithmetical proofs, for example, are true in all possible worlds and the contingency of an individual life doesn't undermine that. In any possible world, whatever mind exists will grasp certain fundamental truths, such as basic arithmetical principles.

I don't mean to be facetious, but has anyone actually verified that arithmetical proofs are true in all possible worlds? Have they visited any of these worlds and done so? Where are these worlds? Have we questioned these other minds about this?

I don't mean to be facetious, but has anyone actually verified that arithmetical proofs are true in all possible worlds? Have they visited any of these worlds and done so? Where are these worlds? Have we questioned these other minds about this? — charles ferraro

Not at all facetious, but the point is, it's true in principle. How could a world hold together where less was greater than more?

Empiricism has its limits. When you ask for 'empirical proof' regarding analytical propositions, then you're crossing those limits. It's like you're asking 'but why does two plus two equal four? You can't prove it.' To which the response is: it need not be proven, it is true according to the axioms of arithmetic. We depend on the structure of rational thought to make sense of the world, to create any kind of theory whatever (pace Immanuel Kant). So if you demand what is the proof for those structures, you're sawing off the limb on which your argument rests.

I believe there was a time when it was thought that someone would be "crossing the line" if they asked the "foolish" question as to whether, or not, it could be empirically verified that Euclidean geometry was applicable to the physical world we experience. I mean, after all, hadn't Kant demonstrated that this had to be the case?

I'm sure, even then, there were those persons who argued vehemently that it was true "in principle" that only Euclidian geometry could apply to the physical world.

But then, lo and behold, the purportedly "inviolable" Newtonian paradigm shifted to the Einsteinian and, as a direct consequence, it was proven empirically, through several rigorous experiments, that the physical world we experience obeyed, instead, a form of non-Euclidean geometry.

Thank heavens empirical science always encourages investigators to challenge and progress beyond the purportedly inviolable tenets and limits set by the current theory.

Who knows what impossibilities the next paradigm shift will make possible?

It really doesn’t have any bearing on the discussion. Einstein relied wholly and solely on reason to establish principles which couldn’t even be verified empirically at the time and in many cases not for decades afterwards.

Furthermore, nothing Einstein said invalidated Newton. It simply showed that Newtonian laws have a limited range of applicability.

Absolutely it has a bearing.

It's simply dead wrong to assert that Einstein relied wholly and solely on reason.

He also relied on the experimental results of physicists who preceded him and he was known for performing mental experiments, but he also eagerly awaited and valued the results of experiments performed by contemporaries to test predictive hypotheses generated by his theories.

He was alive when some major experiments were performed; e,g., when the bending of light rays during a total solar eclipse was verified. Nuclear fission verified his famous equation.

No theory created by human thought has had a perpetual lock on truth. In this sense, I think every theory is subject to invalidation. But this, I submit, is simply an issue of semantics

[our] creative act is open, at every moment, to the possibility of complete cessation. — charles ferraro

The word itself is dead before it is brought alive into time and context--pedestrian, mundane, banal, or contrary, unexpected, mad. Our very expressions begin and end. But, even in sight of its death, some writing holds itself responsible as we are when we speak: to answer for our expression, stand by it, be seen in it, to make ourselves intelligible, known.

If we turn Descartes around (as Plato, Heidegger, and Wittgenstein suggest for ourselves) he is not as worried about "existence" as knowledge, worried about its certainty to ensure his world (a piece of wax) and even himself.

So we can hope to find a kind of knowledge (thought) that will take the responsibility for ourself away, or we can ensure that we are known by our thoughts. We can look for certainty, or we can be certain, specific, thorough, diligent, resolute.

this perpetual openness to and oppressive, arbitrary, unrelenting subjection to the possibility of complete cessation clearly indicates, to me, that the contingent Cartesian thinking and the indubitably certain contingent Cartesian existing don't really matter that much, even if they are man's own creation. — charles ferraro

Our desire for a certainty in knowledge kills what it seeks before it begins. Emerson suggests we live open in front. If we are to let ourselves matter, is it our being subject to doubt at every turn that stops our first step? or will we exist only so far as we know?

Yes, Einstein took into account observations, and relied heavily on thought experiments. But he had no data for many of the predictions that were made by theories, which were, in that sense, the product of reason before observation. Reason plays a critical role in Einstein's thought, as in all science generally. What I'm arguing is that basic principles of thought and logic, such as arithmetic proofs and logical laws, are not contingent - they're what philosophy calls 'metaphysical primitives' or 'basic facts', which cannot or need not be explained with recourse to empirical demonstrations. They are assumed to be functional for the purpose of any reasoned argument. This is not 'a theory' but a principle, and I don't believe you've shown it not to hold.

↪jgill
Presumably you think Newton's most important contribution was the sterling job he did as head of the Royal Mint — Bartricks

I'm humbled by your perceptive reply. :roll:

It's like you're asking 'but why does two plus two equal four? You can't prove it.' — Wayfarer

I can. Let's use apples, no, oranges. No, wait - apples.

Take two apples - put them in an empty bag.
Take two apples - put them in the same bag.
How many apples are in the bag?

It's not a mere convention that there are four apples in the bag.

One could use a different word to represent the number four, but the integer exists in reality - insofar as the universe is not considered a whole, and everything in it an indivisible part of that whole, individual objects exist - so numbers exist, and because one object and one object is two objects, then 2+2=4.

Fair enough, and of course I agree, but the point was that there is no need to prove such simple facts. And I’m a Platonic realist about numbers, so I also agree that the integers are real.

Fair enough, and of course I agree, but the point was that there is no need to prove such simple facts. And I’m a Platonic realist about numbers, so I also agree that the integers are real. — Wayfarer

I was just looking to prove that 2+2=4, but it seems like I've cracked open a whole other can of worms. I've read a couple of short essays, and I'm not sure now where I stand on the question of whether numbers are real. At the very least, there is a question - as to what a number actually is, and in what sense it can be said to exist? Apples isn't the answer - because apples are a number of objects, as opposed to numbers as objects in themselves.

Platonism assumes the existence of ideal forms - but you qualify that with realism? So, what - no ideal forms for you?

Hey there’s a really good current essay on this subject....quick google....here, I’ve been meaning to start a thread on it, perhaps I will, it’s a bit tangential to this one, but have a read.

because one object and one object is two objects, then 2+2=4. — counterpunch

Watch raindrops, sliding down a window.

One plus one equals one.

Watch raindrops, sliding down a window. One plus one equals one. — Banno

Twice the size!

Hey there’s a really good current essay on this subject....quick google....here, I’ve been meaning to start a thread on it, perhaps I will, it’s a bit tangential to this one, but have a read. — Wayfarer

I read the essay, and the answer came to me. Numbers exist in the relations between objects. One apple is one apple. One apple and one apple is two apples. The number exists in the relation of one apple to another; not in the apple itself, not in some platonic ideal beyond time and space - but in the relation between objects. It's too simple, too obvious. Why ain't it that?

"Thinking about geometric objects is perhaps the clearest way to think about abstract objects. A line segment (a true, geometric line segment) is a perfectly straight, one-dimensional object with a determinate length. There are no such objects in space-time. Every object you could possibly interact with is three-dimensional — no matter how thin a piece of, say, plastic you create, it always has a height and a thickness, giving it three dimensions. Nothing, therefore, in the concrete world, is a real geometric line segment. We have things that approximate line segments — very straight, very thin objects. But none of those things will ever be perfectly straight and with zero thickness. So if there does, somehow, exist a true line segment, it certainly isn’t in the concrete world, and therefore it must be in some sort of abstract realm."

Neither does a unicorn, or a griffin exist in the concrete world - yet we can imagine them, and draw them - because we know what a horse looks like, or what a lion and a eagle look like, and can conflate them. So, perhaps a true geometric line segment is like that - an abstraction from the relations between real objects; then the idea of numbers as existing in the relations between objects holds up.

Perfectly true - but only a rational mind can see it.

If you were able to somehow go back in time, and map every mathematical thought in chronological order, surely it would begin with the relationship of one apple to one more apple, and get increasingly abstract.

Also, it's said that phylogeny recapitulates ontogeny - and you don't start kids out learning math with abstract objects. You begin with one apple and one more apple - and build toward abstract objects. So, all in all - maybe this is a wrong end of the telescope problem.

Maybe numbers do exist, primarily in the relations between objects. Then, expert mathematicians forget their infant education, and talk about numbers existing as abstract objects, and cite highly derived ideas, like geometric line segments and set theory, as proof of an abstract realm created through logical manipulation of those basic, real numerical relations.

the point is, they’re the same for all who think, and they can be used to encode ideas which are universally applicable. So, they’re real. But they don’t exist - there are no numbers ‘out there somewhere’. They’re purely and simply objects of rational intellect, but they’re real. That’s the inconvenient truth for empiricists.

Why would it be a problem for empiricists? If you see two apples and I see two apples - there are, empirically speaking, two apples.

Assuming two exists only as the relation of one apple to the other, it becomes an adjective, and it makes as much sense to say 'two' exists, as it would to say high, or long, or lazy - exist as some abstract ideal.

Descartes' use of hyperbolic doubt and, in particular, the constant deception caused by the "evil genius," postulated conditions under which it would be possible to conceive that even the necessary "a priori" truths of logic and mathematics, his "clear" and "distinct" ideas, would be false in this world and in all possible worlds. That's why he desperately, and I think unsuccessfully, tried to argue for the existence of a "good" Deity that would, nevertheless, guarantee their truth.

Thus, only the Cogito Sum, which survived hyperbolic doubt, would be an indubitably certain first principle in this world and in all possible worlds. But it would still be contingent in this world and in all possible worlds.

Contingent upon his existing, for sure.

In other words, for any human who performs the Cogito Sum in the first person, present tense mode, be it performed either in this world or in any possible world, the Cogito Sum would always be both indubitably certain and contingent; not so much in the sense of dependency, but in the specific sense that both the Cogito (I think), and the Cogito generated Sum (I am), would always be experienced as being open and vulnerable to the possibility of complete cessation.

Sure. But I still don’t see why the possibility of cessation is a consideration. Descartes never set out to prove the immortality of the soul, only an indubitable foundation for thought.

Precisely because contingency is an essential characteristic exhibited by the human Cogito Sum that is equal in importance to its indubitable certainty and, therefore, should not be dismissed out of hand.

Also, I think the notions of contingency and necessity, as I defined them, play a crucial role in understanding whether, or not, Descartes' Ontological Argument for the Existence of God works.

I am taking the liberty of forwarding to you a piece I wrote to verify this contention. It's lengthy, but I hope you will enjoy it.

CRITIQUE OF DESCARTES’ ONTOLOGICAL ARGUMENT FOR THE EXISTENCE OF GOD

That the existence of God may be rightly demonstrated from the fact that the necessity of His existence is comprehended in the conception which we have of Him.

Rene Descartes

The (ontological) argument does not, to a modern mind, seem very convincing, but it is easier to feel convinced that it must be fallacious than it is to find out precisely where the fallacy lies

Bertrand Russell

It is this author’s contention that Renee Descartes should have rejected the validity of all ontological arguments for the existence of God and that his philosophy would have provided him with a unique and sound rationale for explaining why such arguments had to be false. Descartes should have realized that his version of the ontological argument, as well as the version formulated before him by Anselm, was simply incompatible with the new philosophical methodology and criteria he established for determining indubitably certain existence.

It was not sufficient for Descartes and Anselm before him merely to present the individual with the idea, or definition, of a necessary being and then, by performing a detailed analysis of the idea, or definition, try to claim to have demonstrated successfully the necessary existence of such a being.

I submit that Descartes’ own well-defined methodology and explicit criteria for determining indubitably certain existence should have prompted him, instead, to explain (a) the difference between contingent thinking activity and necessary thinking activity, and (b) the corresponding difference between contingent personal existence and necessary personal existence. The specific definition of the terms contingent and necessary, as used in this paper, will be made clear during the following discussion.

In Meditation II, Descartes presented the reader with a detailed explanation of the human Cogito Sum along with the method the reader could use to realize it. He claimed that a person attempting to doubt his own existence, even under the most extreme (hyperbolic) of scenarios (the dreaming doubt and the malicious demon doubt), would ultimately and inevitably realize or intuit, during his doubting activity, that his existence was an indubitably certain existence. A simultaneous intuition or realization would occur that not existing while doubting or thinking was impossible for the thinker. Or, phrasing it positively, a simultaneous intuition or realization would occur that existing while doubting or thinking was indubitably certain for the thinker. As Descartes put it: “I am, I exist. This is certain. How often? As often as I think.”

However, Descartes did not say that his existence was necessary-in-itself. He said only if, and when, he doubted, only if, and when, he thought, only then, during the time of their occurrence, did he simultaneously intuit his existence to be indubitably certain. If he ceased to think for an instant of time, then Descartes claimed that he would have no ground for believing that he could have existed during that instant. As Descartes cautioned: “For it might indeed be that if I entirely ceased to think, I should thereupon altogether cease to exist.”

So, then, according to Descartes, a person’s thinking activity is contingent in the specific sense that it is experienced by the person as always being open to the possibility of complete cessation and non-existence. In other words, the Cogito portion of the Cogito Sum is experienced by the person, in the first person, present tense mode, to be contingent thinking activity (a contingent Cogito), since it is experienced as always being open to the possibility of complete cessation and non-existence.

Search as one will, there is no separate or concomitant intuition available which would also assure the person, beyond all reasonable and hyperbolic doubt, that his doubting or thinking is an activity impervious to the possibility of complete cessation and non-existence. And the force of this realization would apply equally to all the many different modes of the person’s thinking activity such as perceiving, inferring, deducing, imagining, remembering, conceiving, speculating, calculating, hypothesizing, etc.

Descartes showed how the performance of a human Cogito Sum did, in fact, yield the intuition of an indubitably certain, yet contingent, personal existence (the contingent human Sum) based upon, emerging from, and restricted to the human person’s simultaneous experience of the occurrence of its contingent thinking activity (the contingent human Cogito). Or, stating it more succinctly, a person’s contingent thinking activity (the human Cogito), during the time that it is experienced by the person, always provides the person with a simultaneous intuition of the indubitable certainty of that person’s contingent personal existence (the human Sum).

Surprisingly, in none of his subsequent meditations did Descartes attempt to present the reader with a detailed explanation of the divine Cogito Sum which would have paralleled nicely the detailed explanation of the human Cogito Sum he offered in Meditation II.

Preoccupied as he was with the urgent need to provide a divine guarantee for his clear and distinct perception criterion of truth, in Meditation III Descartes decided to present the reader with a series of more, or less, traditional a posteriori arguments for the existence of God and, in Meditation V, he decided to present the reader with his a priori ontological argument for the existence of God based, curiously enough, upon his clear and distinct perception criterion of truth.

Nevertheless, had he intended to do so we suspect Descartes could have provided a detailed explanation of the divine Cogito Sum along the following lines.

If one assumes the divinity thinks, then its thinking activity (the divine Cogito) would be necessary in the specific sense that it would be experienced by the divinity as always being closed to the possibility of complete cessation and non-existence and, as such, it would always provide the divinity with an intuition of its indubitably certain necessary personal existence (the divine Sum).

In other words, he could have explained how the performance of a divine Cogito Sum would have provided an intuition of indubitably certain necessary personal existence (the divine Sum) based upon, emerging from, and restricted to the divine person’s experience of the occurrence of its necessary thinking activity (the divine Cogito).

The divine person’s necessary thinking activity (the divine Cogito) would provide the divine person with an intuition of the indubitable certainty of the divine person’s necessary personal existence (the divine Sum).

He could have gone on to explain that IF the human person were also able to experience the occurrence of such necessary thinking activity (the divine Cogito), then the human person, too, would be able to experience it as always being closed to the possibility of complete cessation and non-existence.

But that since the human person is, in fact, simply not able to experience the occurrence of necessary thinking activity (the divine Cogito), in the same way as the human person is able to experience the occurrence of contingent thinking activity (the human Cogito), the human person is, therefore, prohibited from ever having direct access to an intuition of indubitably certain necessary personal existence (the divine Sum).

This Cartesian-based distinction between the impossibility of having a personal experience of necessary thinking activity and the possibility of having a personal experience of contingent thinking activity should not be confused with the traditional distinction between an essence that contains within itself the reason for its existence (necessary being) and an essence that does not contain within itself the reason for its existence (contingent being). The Cartesian-based distinction is grounded in, and can be verified through, a person’s experience, whereas the traditional distinction is grounded in a person’s abstract thinking but cannot be verified through a person’s experience.

From a Cartesian-based perspective, the central issue is the possibility of having a personal experience of thinking activity that can cease to occur and can go out of existence versus the impossibility of having a personal experience of thinking activity that can never cease to occur and can never go out of existence.

Human thinking activity is contingent being because the human person experiences his thinking activity can cease to occur and can go out of existence – nothing more, nothing less. The human person’s, alone, is the I think contingently, I exist contingently (Cogito contingenter, Sum contingenter).

By contrast, divine thinking activity is necessary being because the divine person experiences that its thinking activity can never cease to occur and can never go out of existence - nothing more, nothing less. God’s, alone, is the I think necessarily, I exist necessarily (Cogito necessario, Sum necessario).

It is simply impossible for a human being to have a personal experience of thinking activity that can never cease to occur and can never go out of existence (the divine Cogito).

However, from a Cartesian perspective, it is precisely this impossible experience which is the indispensable prerequisite that would enable a human being to have a performative intuition of the indubitable certainty of necessary personal existence (the divine Sum), i.e., the existence of God.

But, unfortunately, all ontological arguments lack this indispensable experiential prerequisite. And, in response to Russell, this is precisely where the fallacy of the ontological argument lies!

For whatever reasons, the preceding line of thought is what Descartes chose neither to pursue, nor to explain.

Nevertheless, from a Cartesian point of view based upon a well-defined Cartesian methodology and explicit criteria for determining indubitably certain existence, I would submit (a) that the occurrence of necessary thinking activity (the divine Cogito) is precisely what a person would have to be able to experience in order to make a legitimate claim to having an intuition of indubitably certain necessary personal existence (the divine Sum), and (b) that this Cartesian-based explanation of what would be required for a human person to successfully execute an intuition of indubitably certain necessary personal existence (the divine Sum) is far superior to Descartes’ ontological argument and that of his predecessor, Anselm.

This Cartesian-based critique specifies precisely what is fallacious about Descartes’ ontological argument, Anselm’s ontological argument, and all other ontological arguments for the existence of God in a manner uniquely different than the critiques proposed by St. Thomas Aquinas, David Hume, Immanuel Kant, Arthur Schopenhauer, and Gottlob Frege.

Ontological arguments, being conceptually abstract through and through and remaining completely detached and isolated from the empirical realm, lack the requisite foundation of a personal human experience of necessary thinking activity (the divine Cogito). Only the possibility of having such a personal experience would also permit a human person to have an intuition of indubitably certain necessary personal existence (the divine Sum).

It is of interest to note, too, that all the critiques cited above are essentially as conceptually abstract as the ontological arguments they seek to contest. The total inability of a person to experience the occurrence of necessary thinking activity is never made the central issue of contention. For all these critics, the perennially unresolved central issue is simply the logical validity, or invalidity, of the abstract reasoning involved in the ontological arguments. Without exception, this is their exclusive, limited focus.

I submit that the Cartesian-based critique succeeds in altering this traditional focus since it offers a unique, experientially grounded explanation for why, ab initio, all ontological arguments for the existence of God must be false.

Certain assumptions shared by Descartes’ arguments for the existence of God, be the arguments a posteriori or a priori, are that the ideas of the infinite and the perfect are ontologically prior to the ideas of the finite and the imperfect, and that the ideas of the infinite and the perfect are innate to the human mind because they are implanted there by God.

For example, for Descartes my idea that I think contingently (which is my idea of a finite and imperfect activity) presupposes an ontologically prior, innate idea of what it means to think necessarily (which is my innate idea of an infinite and perfect activity).

Or, to understand that I think contingently (a finite and imperfect activity) requires that I must have some ontologically prior, innate understanding of what it means to think necessarily (an infinite and perfect activity). However, as this line of reasoning relates to the central theme of this essay, I would submit, contrary to Descartes’ position, that my understanding of the idea of necessary thinking activity (an infinite and perfect activity) is not innate to the human mind and is not implanted there by God.

Neither is the idea of my contingent thinking activity (a finite and imperfect activity) obtained, as Descartes would claim, by my limiting or bounding, in some way, the ontologically prior, innate idea of necessary thinking activity (an infinite and perfect activity). Instead, my idea of necessary thinking activity is a direct result of my deliberate attempt to try to remove, albeit unsuccessfully, that characteristic from the idea of my contingent thinking activity which limits and constrains it; viz., its vulnerability to the possibility of complete cessation and non-existence. This, I submit, is the genuine way in which I arrive at an understanding of the idea of necessary thinking activity (an infinite and perfect activity).

Nevertheless, it does not necessarily follow, either from the former interpretation of Descartes or from the latter interpretation of this author, that I can have a direct personal experience of necessary thinking activity (an infinite and perfect activity) in the same way as I do, in fact, have a direct personal experience of contingent thinking activity (a finite and imperfect activity).

As I see it, the central issue is not a matter of the possibility of my being able to have, or not to have, an idea of perfect thinking activity or an idea of perfect being – be those ideas innate, adventitious, or factitious. Instead, the central issue is a matter of the possibility of my being able to have, or not to have, a direct personal experience of that perfect thinking activity or of that perfect being.

Or, approaching it from a slightly different direction, doubts and desires may come from an understanding that I lack something, and that I would not be aware of that lack unless I was aware of a more perfect being that has those things which I lack. However, my ability to have an idea of, or conception of, or understanding of, or awareness of a more perfect, or infinite, being that possesses all those things which I lack (inclusive of necessary thinking activity), does not mean that I am also able to have a direct personal experience of that being and its necessary thinking activity in precisely the same way as I am able to have a direct personal experience of my being and my contingent thinking activity.

Certainly, I can postulate the existence of a being that thinks necessarily and exists necessarily, but I cannot have a direct personal experience of the necessary thinking activity which would simultaneously yield an intuition of the indubitably certain existence of such a necessary being. Again, I can perform the “Cogito contingenter, Sum contingenter,” but I cannot perform the “Cogito necessario, Sum necessario.”

Descartes’ a priori ontological argument for the existence of God is not an experientially grounded performative argument like the one he formulated that successfully and persuasively proved the existence of the human self. His ontological argument, lacking the crucial, indispensable experiential foundation of necessary thinking activity, is destined to fail from its very inception. It is a non-persuasive, quasi-intuitive argument espousing a so-called self-validating idea of God which is given in consciousness and which represents God as existing, but which, in fact, completely misses the mark.

In fact, one could assert even further that the ultimate test of the efficacy of any argument for the existence of God, be that argument a priori or a posteriori, does not consist in the ability of that argument to provide the meditator with a clear and distinct idea of God’s necessary personal existence. Instead, one could assert that the efficacy of any such argument is determined, first and foremost, by whether, or not, it can engender in the meditator a direct personal experience of necessary thinking activity (the divine Cogito). And even assuming such an argument can engender in the meditator a direct personal experience of necessary thinking activity, then can it also engender in that meditator a simultaneous intuition of indubitably certain necessary personal existence (the divine Sum)?

However, in conclusion, this author knows of no traditional, professionally recognized, a priori or a posteriori argument for the existence of God that has succeeded in providing the meditator with the requisite foundation of a direct personal experience of necessary thinking activity (the divine Cogito) while also engendering in the meditator a simultaneous intuition of indubitably certain necessary personal existence (the divine Sum).