The Platonic Mathematical World

[Zarathustra]

I don't think we have innate knowledge, but we do have innate structures of our consciousness/experience/understanding that have to be "present" in order to have any consciousness/experience/understanding at all. For instance, the two structures I gave above: there is no consciousness without an object of consciousness (even if that object is consciousness itself, e.g. self-consciousness). And there's no consciousness without intentionality, i.e. being directed to an object in a particular way (whether that's emotive, cognitive, perceptual, etc.).

Kant talked about necessary structures of experience like time and space ... not the physical properties, but temporality and spaciality themselves as the "field" of perceptual experiences (regardless of whether or not those perceptions are "of a world"), without which perceptual experiences could not exist or occur at all. Even dreams have spacial and temporal qualities--or they are "framed" spatially/temporally. Maybe time doesn't flow straight or smoothly in dreams, and maybe space doesn't conform to Euclidean geometry, but we can't have a perceptual experience at all without some form of spatially/temporally. These are a priori structures in that sense.

And one could argue for other such structures of experience, or knowledge. Many philosophers have. "Substance," for instance, is a concept that is supposedly not given in experience, if you think of perceptual experience as entirely mental. We never perceive substance in itself, and yet our perceptual experiences of the world wouldn't "adhere together" without this concept "anchoring" them. We apparently intuit it this concept as a necessary, a priori condition of having any perception of a physical world whatsoever. Etc.

[SerScot]

Mighara Sovmadhi,

Is higher mathematics a result of deductive reasoning? Or inductive reasoning?

Mostly deduction, but also some more "constructive" but non-inductive/abductive (sometimes the word is "ampliative") reasoning. Good luck understanding the latter (we philosophers/mathematicians don't know what to make of it so much, yet, and some of us end up saying that it's "intellectual intuition"!).

"Intellectual intuition". That's really interesting. I saw a description of how to do "Magic boxes" mathematically in a gift we bought for my children over Christmas. It was really interesting and very intuitive.

Could "intellectual intuition" be an anticipation of something that is a priori that we haven't figured out yet?

Avatar,

I feel that anything inductive can't be an a priori truth, because it isn't based on experience (ie knowledge) but on assumption.

But Mathematics, that has been rigorously proven, is the opposite of induction. It has been shown it cannot be otherwise. If it cannot be otherwise it is, by definition, objective and deductive. If it is objective and deductive it is an a priori truth.

[Hashi Lebwohl]

But Mathematics, that has been rigorously proven, is the opposite of induction. It has been shown it cannot be otherwise. If it cannot be otherwise it is, by definition, objective and deductive. If it is objective and deductive it is an a priori truth.

I don't know if I would go that far with it. Mathematics did not exist before humans were around to invent it; however, since inventing it we have developed an internally-consistent framework that withstands independent investigation and verification. In nearly 10,000 years of human history, no one has come up with an alternative to 1 + 1 = 2 or "all rays originating at the focus and reflecting off a parabola result in parallel lines".

The system of mathematics we have also conforms to observable reality but it doesn't define observable reality. The second derivative at a point of inflection is not zero because we say it is; we equate it to zero because that is what we observe about the nature and behavior of graphs.

[Mighara Sovmadhi]

Could "intellectual intuition" be an anticipation of something that is a priori that we haven't figured out yet?

Originally, to my knowledge, "intellectual intuition" was a sort of technical Kantian phrase. In Kant, an intuition (in his language aunschaung or something) is not a feeling or a hunch, it is a conscious state with particular content or attributes. It is, that is, particular consciousness. This subsumes sense perception (percepts are particulars) but leaves room for non-sensory intuition, which would be what intellectual intuition is supposed to be (Kant asserts this of the hypothetical absolute God of his).

Kant more or less denied that we can intellectually intuit things, but his phrase has been appropriated by those who either deny his noumena-phenomena/appearances-dingansich dichotomies, or think there's at least a little bridge between such categories, or who don't know what he meant by his own phrase.

[Mighara Sovmadhi]

The second derivative at a point of inflection is not zero because we say it is; we equate it to zero because that is what we observe about the nature and behavior of graphs.

Perhaps it seems that way to non-mathematicians, but we mathematicians don't think of it like this. The visible graph is a tool we use to convey our understanding of a function to others, but the actual function in our minds is only tangentially like the lines we draw on pages (or computer screens). For starters, for instance, graph lines are not infinitely thin, but the pure line of a function corresponding to a graph would be...

[Mighara Sovmadhi]

Though he contributed no theorems of which I am aware to mathematical history, Kant does describe the mathematical state of mind (as entered into when doing proofs) very well here:

The construction of a conception is the presentation a priori of the intuition which corresponds to the conception. For this purpose a non-empirical intuition is requisite, which, as an intuition, is an individual object; while, as the construction of a conception (a general representation), it must be seen to be universally valid for all the possible intuitions which rank under that conception. Thus I construct a triangle, by the presentation of the object which corresponds to this conception, either by mere imagination, in pure intuition, or upon paper, in empirical intuition, in both cases completely a priori, without borrowing the type of that figure from any experience. The individual figure drawn upon paper is empirical; but it serves, notwithstanding, to indicate the conception, even in its universality, because in this empirical intuition we keep our eye merely on the act of the construction of the conception, and pay no attention to the various modes of determining it, for example, its size, the length of its sides, the size of its angles, these not in the least affecting the essential character of the conception. [Critique of Pure Reason, transcendental doctrine of method: the Discipline of Pure Reason]

[SerScot]

I just heard about a new variation on the Incompleteness therom. A NASA scientist is trying to rigorously prove with formal logic that it is impossible to know everything in the Universe and still be within the Universe to interact with it:

www.scientificamerican.com/article/limi ... rehension/

From the article:

Wolpert proves that in any such system of universes, quantities exist that cannot be ascertained by any inference device inside the system. Thus, the “demon” hypothesized by Pierre-Simon Laplace in the early 1800s (give the demon the exact positions and velocities of every particle in the universe, and it will compute the future state of the universe) is stymied if the demon must be a part of the universe.

Researchers have proved results about the incomputability of specific physical systems before. Wolpert points out that his result is far more general, in that it makes virtually no assumptions about the laws of physics and it requires no limits on the computational power of the inference device other than it must exist within the universe in question. In addition, the result applies not only to predictions of a physical system’s future state but also to observations of a present state and examining a record of a past state.

The theorem’s proof, similar to the results of Gödel’s incompleteness theorem and Turing’s halting problem, relies on a variant of the liar’s paradox—ask Laplace’s demon to predict the following yes/no fact about the future state of the universe: “Will the universe not be one in which your answer to this question is yes?” For the demon, seeking a true yes/no answer is like trying to determine the truth of “This statement is false.” Knowing the exact current state of the entire universe, knowing all the laws governing the universe and having unlimited computing power is no help to the demon in saying truthfully what its answer will be.

I've always believed the saying "The Universe is not only stranger than we imagine, it is stranger than we can imagine" was true. I may have been more right than I knew.

[Hashi Lebwohl]

I found the original published paper in the archives at Cornell; it is a 43-page paper which should make for a decent evening of reading. Be aware--it gets deep.

Section 6.2 deals with the philosophical implications for those who are interested.

Return now to the case where U is a set of laws of physics (i.e., the set of all worldlines consistent with a set of such laws). The results of this subsection provide general restrictions that must relate any devices in such a universe, regardless of the detailed nature of the laws of that universe. In particular, these results would have to be obeyed by all universes in a multiverse [27,28,29]. Accordingly, it is interesting to consider these results from an informal philosophical perspective. Say we have a device C in a reality that is outside distinguishable. Such a device can be viewed as having “free will”, in that the way the other device s are set up does not restrict how C can be set up. Under this interpretation, Thm. 1 means that if two devices both have free will, then they cannot predict/recall/observe each other with guaranteed complete accuracy. A reality can have at most one of its devices that has free will and can predict/recall/observe the other devices in that reality with guaranteed complete accuracy. (Similar conclusions hold for whether the devices can “control” each other; see Sec. 7 below.)

Thm. 3 then goes further and considers devices that can emulate each other. It shows that independent of concerns of free will, no two devices can unerringly emulate each other. (In other words, no reality can have more than one universal device.) Somewhat tongue in cheek, taken together, these results could be called a “monotheism theorem”.

Now suppose that the domain of a reality is a set of worldlines extending across time, and consider “physical” devices that are identified with system s evolving in time. (See discussion just after Def. 7.) Prop. 5 tells us that any universal device must be infinite (have infinite X (U)) if there are other devices in the reality that are copies of it. Since the time-translation of a physical device is a copy of that device, this means any physical device that is ever universal must be infinite. In addition, the impossibility of multiple universal devices in a reality means that if any physical device is universal, it can only be so at one moment in time. (Its time-translation cannot be universal.) Again somewhat tongue in cheek, taken together this second set of results couldbe called an “intelligent design theorem”. (See Sec. 7 for related limitations concerning devices that are used to control one another.)

[Vraith]

I'm gonna have to dive into that sometime soon.
It seems to gel with things I've thought for a very long time [though of course I haven't looked at it with true rigor...I probably don't even have the capacity to look at it that way].
I think the first time I started moving that way was because in some SF book way back when I was in high school [and was probably written even before that, since my school library had little SF, even less "new" SF] some character espoused the opinion that at the moment a computer became truly intelligent it would find itself with a subconscious just like us. Would no longer be able to fully understand its own nature, its own functions, its own goals/desires/purposes.
Years later, finding GEB added much to the tendency.

[SerScot]

Vraith,

Do you think the "Halting problem" and "non-computablity problem" for any Universal Computer creates a serious problem for proponents of "Strong AI"? After all even Turing's test only tests the ability of something to mimic human speech, it doesn't test for real consciousness.

[Vraith]

Vraith,

Do you think the "Halting problem" and "non-computablity problem" for any Universal Computer creates a serious problem for proponents of "Strong AI"? After all even Turing's test only tests the ability of something to mimic human speech, it doesn't test for real consciousness.

No, I don't...as far as I understand those problems. I think strong AI is easily [heh...relatively speaking] achievable.
I wouldn't be shocked if it happened within 25 years, perhaps sooner, I WOULD be shocked if it DIDN'T happen within 50.
There are a couple things I think make it not only likely but practically guaranteed [barring catastrophic something happening to us soon].
Fuzzy Logic is one, on the programming side.
Many on the physical machine...quantum, for instance.
And a "mix" of the two is content-addressable memory and massive parallel processing [which may or may not require fuzzy logic software or some other new method/code type in order to run].
This isn't my field, or anything like it, so I'm sure there are a lot more ideas/possibilities out there amongst those for whom it IS their talent and expertise.

And I think of Strong AI somewhat differently than perhaps is technically accurate. It is possible that, if the definition is perform any and all intellectual tasks a human can it might not happen. We might keep some human-only capacities. BUT the machine will surely be able to perform intellectual tasks that WE can't, too. [I don't just mean speed, accuracy, volume...I mean think thoughts, have ideas, gather/create knowledge that we are incapable of because of our brain's mechanical design/build]

[Avatar]

Agree with Vraith...it's coming.

--A

[SerScot]

According to Penrose it may not be possible.

[Vraith]

According to Penrose it may not be possible.

Yes, but...unless he's come up with a new argument...the main problem he has with it is:
There are things our minds do that are NOT based on algorithms.
BUT all the ways we can make machines run IS based on algorithms.
So there are and always will be problems the machine can't really even "think about," let alone solve...

He has to be right about both the claims...but what if he's not?
Just for instance...what if our minds ARE algorithm-based, but we get around the shortcoming by changing viewpoints? We have an algorithm for shifting algorithms [or using multiple ones] when we run into a halt or non-computable.
It's not at all uncommon for a problem to be solved by just using a different math system.
[[FWIW, I don't think our minds are entirely algorithm based...maybe our future software won't be either. Fuzzy logic is a step in that direction, or a different way of doing the same thing.]]

[SerScot]

Vraith,

Don't you have an obligation to show that organic minds are actually algorithm based? Doesn't this also create problems for Strong AI when we consider whether P=/=NP. If polynomial time and non-polynomial time are not equal couldn't we end up with serious halting problems when an AI seeks to answer a non-polynomial problem?

[Zarathustra]

I just heard about a new variation on the Incompleteness therom. A NASA scientist is trying to rigorously prove with formal logic that it is impossible to know everything in the Universe and still be within the Universe to interact with it:

www.scientificamerican.com/article/limi ... rehension/

From the article:

Wolpert proves that in any such system of universes, quantities exist that cannot be ascertained by any inference device inside the system. Thus, the “demon” hypothesized by Pierre-Simon Laplace in the early 1800s (give the demon the exact positions and velocities of every particle in the universe, and it will compute the future state of the universe) is stymied if the demon must be a part of the universe.

Researchers have proved results about the incomputability of specific physical systems before. Wolpert points out that his result is far more general, in that it makes virtually no assumptions about the laws of physics and it requires no limits on the computational power of the inference device other than it must exist within the universe in question. In addition, the result applies not only to predictions of a physical system’s future state but also to observations of a present state and examining a record of a past state.

The theorem’s proof, similar to the results of Gödel’s incompleteness theorem and Turing’s halting problem, relies on a variant of the liar’s paradox—ask Laplace’s demon to predict the following yes/no fact about the future state of the universe: “Will the universe not be one in which your answer to this question is yes?” For the demon, seeking a true yes/no answer is like trying to determine the truth of “This statement is false.” Knowing the exact current state of the entire universe, knowing all the laws governing the universe and having unlimited computing power is no help to the demon in saying truthfully what its answer will be.

I've always believed the saying "The Universe is not only stranger than we imagine, it is stranger than we can imagine" was true. I may have been more right than I knew.

And yet, not only can we imagine such a universe where the above proof can be true, we can construct such a proof, demonstrate it, and understand it ... even though we're inside the universe. In some sense we can see a truth "bigger" than what Laplace's demon could see or prove. So in constructing such proofs, we're in some sense getting beyond the very limitation which the proof points out. We're seeing the "end result" of a non-halting problem.

... the moment a computer became truly intelligent it would find itself with a subconscious just like us. Would no longer be able to fully understand its own nature, its own functions, its own goals/desires/purposes.

That's an interesting concept. It's like self-consciousness itself "reverbates" with recursive levels below the immediate, where one is generally aware of the world, then less immediately aware of one's own awareness of the world, and even less immediately aware of one's own awareness of one's awareness of the world ... and so on.

I was just rereading some Heidegger today (Being and Time) where he talks about a conscious human being (Dasein) as being "ontically nearest" and yet "ontologically farthest." That is, in factual terms of our actual existence, we are the closest being to ourselves (because we ARE ourselves). But it is this very closeness which often prevents us from properly understanding our being in a more general sense, because the every-day immediacy of our experience can outshine our awareness of the more general structures of our existence, all the possible ways in which we could currently exist (kind of like the sum of possible "worldlines" in Hashi's quoted article). Those ontological structures are very real, we embodied them each moment in our particular form of existence, as occupying one instance of all those possibilities. And yet the immediacy of the actual can blind us the reality of the general/possible.

However, this doesn't mean we can't reorient our ordinary, every-day focus away from the immediate to our participation in these ontological possibilities (as an instantiation of them). And a truly self-conscious computer would likely be able to do the same.

So that's a long-winded way of saying that I agree with the basic concept of the character's opinion which you've referenced. Though I wouldn't describe it as "subconsciousness," I think a self-conscious computer would have these levels of greater/less awareness, simply because of the nature of being a self-conscious entity in a world. It's an inherent part of the structure of such a being.

[SerScot]

Zarathustra,

Our own emotions and emotionality prevent us from objectively knowing what we should know best... ourselves, because those emotions push us into subjectivity?

[Zarathustra]

Emotions can certainly distract us, so in that sense they can perform the same role. But emotion can also focus one in the moment, even expand one's awareness "of the moment" to include the "infinite." Or in another sense, emotions can connect us with others or with the world, beyond our own subjectivity. In fact, Heidegger characterizes our being-in-the-world in terms of "care," being personally invested with, concerned with objects in the world.

And besides, what I'm talking about can be done purely "conceptually," though phenomenological analysis is perhaps a better term.

[Vraith]

Zarathustra,

Our own emotions and emotionality prevent us from objectively knowing what we should know best... ourselves, because those emotions push us into subjectivity?

I don't think I can go with that/the things it implies. Our emotions are an inextricable and necessary part of our being.
They don't force us into subjectivity...we would be subjective beings even if we were completely without emotions [though we wouldn't be humans, and our line might not have survived long enough to produce us...emotions are positive survival traits, they are selected FOR, at least in this world's environment.

When we create an AI, it will in no way be less subjective than we are.
And I suspect one of the very first things that will happen is it will spontaneously generate/create/become aware of desires/wanting. [ that may be the first sure sign that it IS intelligent, perhaps even the strange attractor/tipping point that flips it INTO true intelligence]

[SerScot]

So… is the Bing Chatbot… an actual strong AI or just a machine mimicing human speach patterns?