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Did math accidentally evolve?

Or are we just connecting to the universe, as the Design of Life authors think.I’ve always found the connection between soft math and useful information easy (like, you get charged for a side of fries you never ordered, and never would have ordered).

But HARD math? That’s about something else for sure. Go here for more about why hard math matters.

Also: Today at the Post-Darwinist

Christianity Today features news item on young astronomer denied tenure

Catholic Darwinists to congregate in Rome?

Also: Today at The Mindful Hack:

The myth of the Christian Right: What happens when you ask Democrats if they too are born again?

God must exist, otherwise he wouldn’t be able to enjoy this debate.

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17 Responses to Did math accidentally evolve?

  1. I missed all but the last five minutes, but Jonathan Wells was interviewed on the radio today by Michael Medved, talk show host and Discovery Institute fellow. I assume he was promoting “The Design of Life”. Couldn’t find the audio online.

  2. The existence itself of mathematics and logics is a true philosophical problem. Indeed, we have to explain two different things:

    a) Why and how do human beings know and develop mathematical principles and theories.

    b) Why and how is the universe apparently built on mathematical laws.

    The two questions are anything but trivial. First of all, mathematics appear to be the only form of science which is not empirical. The nature of mathematical knowledge is much debated, and always has been, by mathematicians themselves, and by philosophers. Personally, I completely agree with Penrose, who is in favour of a platonic conception of mathematics: in other words, mathematical objects and principles are in sone way perceived by the mind, but are not derived from empirical experience.

    Moreover, we have to explain why mathematics, whatever it is and whatever its origin may be, is so good at explaining how “physical” laws work. It is important to remember that many fundamental progresses in physical knowledge, especially in the field of quantum mechanics, have been achieved by using very sophisticated and abstract mathematical instrunents (just an example: the use of the mathenatics of complex numbers in quantum mechanics to reach results which in the end are obviously expressed as real numbers), and the results are often completely counter-imtuitive, from a physical and empirical point of view. And yet, the mathematical results are confirmed by empirical experience, up to an incredible level of precision (quantum mechanics is, as far as I understand, even more powerful in its quantitative predictions than classic newtonian mechanics). That incredible correspondence between mathematics and physics is really a mystery, and the only sensible explanation seems to be that the whole universe has been engineered according to pre-existent mathematical laws, and that in some way our mind appears to have direct access to those laws, independently from empirical experience.

  3. I believe the existance of mathematics isn’t a problem for Atheists because they either accept that it doens’t require a reason to exist or appeal to the Anthropic Principle, which is pretty much the same thing as the former in practice.

    While it is useful to use fundamental laws to prove that Materialists do not have a belief system that’s entirely reason or logic based, (because they accept mathematics as it is reasonlessly, and logic is itself closely related to mathematics if I’m not mistaken) this hole in the Materialistic worldview is normally insufficient in preventing Materialists from being “interllectually fulfilled”.

  4. 4

    Are gender differences in math genius by design?

    There was for some years wailing and gnashing of teeth in the U.S. over the fact that “boys score higher that girls on standardized math tests.” The actual fact is that the MEAN scores of males are higher than those of females. And the reason is that there is a “bulge” in the distribution of male scores, about four standard deviations above the mean (i.e., at genius level). The mean of a distribution is sensitive to outliers. When the very high scores are eliminated, the distributions of male and female math scores are very similar. It is worth noting that I first learned about this from a female member of a National Science Foundation panel addressing the “problem” of improving math education for girls.

    In short, there is prima facie evidence that math genius is gender-related (not gender-specific, for there are women who are math geniuses, just as there are women with pattern baldness).

    Former Harvard president Larry Summers was lynched for remarks in a similar vein. Please don’t chase me with a rope.

  5. Cloud of Unknowing, no chase, no rope.

    Here’s the deal: Generally, women’s achievement curves fit inside men’s.

    More men than women get the Nobel Prize, but more men than women are reviled as mass murderers too.

    I don’t think that a politically correct program would change this situation in a good way. It would just as likely cause more boys to fail miserably as cause more girls to succeed spectacularly.

    I feel badly for Dr. Summers, because he was the victim of a cabal determined to deny reality in these matters, as I have written at the Post-Darwinist.

  6. 6

    gpuccio,

    I posted this in the “Which came first — butterfly or caterpillar?” thread (#27; see also #31):

    Back in 1974, an MIT doctoral student and his advisor investigated the problem of deciding the truth of sentences in the weak monadic second-order theory of one successor. (This is a hamstrung, but interesting, logic in which all sentences can be decided true or false. Godel does not rear his ugly head.) They proved that a logical device known as a Boolean circuit would require at least 10^125 logical elements (gates) to process sentences of length up to 610. The number 10^125 is their estimate of how many protons could be packed into the space occupied by the known universe. (Stockmeyer and Meyer, “Cosmological Lower Bound on the Circuit Complexity of a Small Problem in Logic,” Journal of the ACM, 2002.)

    It’s natural to challenge you to find a human — perhaps an autistic savant — who can consistently decide whether long sentences in this theory are true. To generalize, find people who can reliably solve instances of decidable problems in logic that clearly require computational resources beyond those available to any material computing device in the known universe. A Nobel prize is waiting.

  7. I was wondering where this post would lead. It took a bit to get started.

    I guess I am a bit dissappointed that it went into the philosophy of mathematics. The first link in the post discusses a far more relevant topic to this website. How on earth does the neo-Darwinianist explain the fact that we have far more ability to do math than is beneficial for our survival?

    This brings up another puzzling question about our over-capacity. Infants have a whole mental technology designed to hear and understand speech. At about one year old, the brain dammages itself, eliminating the ability to hear speech sounds that the child has not heard. The presumption is that the brain dammages itself to focus on the important stuff. But my question is why all child brains of certain cultural groups don’t come pre-dammaged to match the sounds that that culture uses. Rather, however, it seems that a caucasian baby raised in a village in India has the same language and hearing capacity of the Indian baby. This smacks of overcapacity, and a universality of humanity that is negatively predicted by the neo-Darwinian hypothesis.

  8. 8

    Denyse,

    Generally, women’s achievement curves fit inside men’s.

    Gentle hint: You might want to say that the women’s curves are generally narrower than the men’s. (So much for referring to women generically as broads.) One distribution cannot fit inside another.

    The word “achievement” seems carefully chosen. Let’s make it clear that if math scores of males were normally distributed, only one in a million scores would be more than four standard deviations above the mean. Approaching this level, there are no substantive differences in distribution of male and female scores. But the number of males with scores about four standard deviations above the mean far exceeds the number we would expect if the distribution of scores were normal. The phenomenon is clearly discrete, and it is implausible that differential nurture of males would manifest in such a fashion.

    A relevant observation is that the ratio of males to females in Savant Syndrome is 5-to-1.

  9. Cloud of Unknowing:

    I am very interested in your discourse. I have read your posts in the previous thread. I am not completely sure to understand your point fully, but I’ll try to give my personal reflections.

    First of all, I think that the main point here could be the difference between computational power and the capacity to “understand” mathematical and logical principles, or just anything else.

    We have been led, in the recent history of science, to believe in theories such as strong AI. In the light of strong AI, any understanding should be based on computational power. Well, I don’t believe that to be so, and I am not the only one.

    Penrose, whom I have already cited, is perhaps the most evident case of a mathematician not believing in strong AI. You are probably aware of his argument, based on Godel’s theorem, in favour of the existence of non algorithmic processes of knowledge in conscious beings.

    I am aware that Penrose has been often criticized (like whoever is trying to suggest new ideas which do not conform to materialistic science dogmatism), but I think he has made a very important point. Although I am not a mathematician or a logician (you are certainly ahead of me on that), I am deeply convinced that any real theory of “meaning” is possible only in the context of consciousness. While computation, both mathematical and logical, can be easily done in algorithms, and therefore mechanically, only a conscious, intelligent being can understand the “meaning” of a computation.

    Meaning is pertaining to representation, and representations take place only in consciousness. Meaning is the byproduct of one of the main characteristics of consciousness, its ability to “detach” itself from what it is observing, to be in a “metalevel” any time it is necessary. That is because consciousness requires a subject, which is the simple obvious point that all strong AI supporters seem to miss.

    It is the metalevel representation realized by the subject vs the object that creates “meaning”. Meaning is, first of all, a feeling of the cognitive appropriatenes of a representation. There is no meaning in objective computations, there is no meaning in machines, unless conscious intelligent agents input and/or recognize it (think, for instance, of Searle’s chinese room example). Godel’s theorem, beyond its mathematical conundrums, is a very strong example of the difference of considering meaning as a simple computational property or, more appropriately, as a conscious appreciation.

    Dembski’s work about specification is, in my opinion, a very powerful, although still initial, tentative to understand the “meaning of meaning” in a mathematical framework, differently from how it is intended in information theory, which is mainly a theory of computational resources. The concept of specification, although sometimes elusive, has both a strong empirical basis and a sound theorical context. If we want to understand something of the phenomena of consciousness, I think we have to go deeper in that direction.

    By the way, I was very intrigued by your blog name. I have that very interesting book in my personal library, although I have read only part of it.

  10. 10
    Cloud of Unknowing

    bFast,

    How on earth does the neo-Darwinianist explain the fact that we have far more ability to do math than is beneficial for our survival?

    Humans apparently are born with a language acquisition device, to use Noam Chomsky’s term, but all evidence is that there is no analogous math acquisition device. The vast majority of four-year- olds speak and understand natural language better than any computer does, and there’s usually no effort in achieving that level of competence. Almost everyone who learns basic arithmetic works relatively hard to do so.

    A related topic is risk perception. It is well established that for most people — even those who can balance their checkbooks — perceived risk is very different from actual risk. It appears that acquisition of basic skills in number processing does not stop people from assessing risk as their hunter-gatherer forbears did. The vast majority of humans form grossly irrational beliefs on gut instinct, and typically become angry when confronted with numbers. For instance, most Americans don’t care to hear that the greatest threat to Americans (especially women and children) is, by any sane calculation, the American male. Most people want to allocate huge resources to defend against what frightens or angers them most, and overlook much greater risks that have become familiar.

    I don’t know how you come by the notion that WE “have far more ability to do math than is beneficial for our survival.” Very few of us are good at math, and the inability of most to assess risk numerically, as necessary in the contemporary world, is itself a threat to our survival.

  11. Cloud of Unknowing:

    I think what bFast means is not that human beings in general have a great mathemathical genius, but rather that the human species is perfectly able, even in a few of its implementations, to develop highly abstract and complex math, while our “darwinian” precursors, chimps if you want, are not.

    I don’t literally agree with bFast that the ability to do math, even in its most abstract forms, is not beneficial to our survival: as I already said, quantum mechanics is possinle only thanks to complex math, and I do think that understanding quantum mechanics has certainly had important consequences on our survival, probably positive if we consider all the aspects. But I can certainly agree with bFast that the human potential for higher math is certainly very, very indirectly connected with the possible effects on survival, and as such it couldn’t certainly be selected by darwinian methods, whatever tha may mean.

    But, sometimes, I have the feeling that we are really speaking of total nonsense. Does anybody really, really believe that random mutations and natural selection based on survival capacities have molded, step by step, the transition from chimps to humans in less than one million years, building the necessary brain structures which would allow man to intuit and understand the abstract principles of mathematics, and gradually discover the stunning fact that the world seems to work upon them?

  12. gpuccio: “Does anybody really, really believe that random mutations and natural selection based on survival capacities have molded, step by step, the transition from chimps to humans in less than one million years, building the necessary brain structures which would allow man to intuit and understand the abstract principles of mathematics, and gradually discover the stunning fact that the world seems to work upon them?”

    This is in itself very hard to believe, but the Darwinist evolutionary psychologists have no problem.

    The “just so” story has a dreadful superficial plausibility to conditioned minds.

    Primitive prehominids had the basic mammalian brain, already of great complexity, with maybe a billion neurons.

    Simple arithmentic ability (counting, adding, subtracting, multiplying, dividing) was strongly selected for. There was also selection for more and more sophisticated analytical and planning capacities for such things as hunting, figuring out what human enemies were planning, etc. etc. This amounted to the beginning of analytical, abstract thought. These selection pressures slowly complexified and enlarged the brain especially the cerebral lobes. This made more and more synaptic connections available for all modes of cognition. At the stage of Homo Sapiens the depth of interconnectional complexity of the nerve net and the associated memory was enough to enable new modes of cognition to “emerge”: mathematical abstract thought. These abilities were not directly selected for, but were potentials enabled by increasing brain complexity.

    I think that in truth this kind of just so story does have a little validity, but no more. This part that is true is that advanced mathematical abilities must require something like the human brain, and that for instance prehominids with a cranial capacity of 500cc, for instance, would be incapable of higher mathematical insights.

    It might be pointed out that examples like certain cases of hydrocephalus seem to show that apparently normal human cognition seems to remain even if large parts of the cerebral lobes are (slowly) destroyed. I don’t have a ready answer to these and other cases except to note that it is clear that the human mind interpenetrates or enables in the physical world through the brain (interactive dualism), and seems to be able to surmount brain malfunctions in certain special circumstances. Ultimately the mind and will and intention behind mathematical abstract thought and insight does not reside in the brain. It expresses through the brain, and except in special circumstances this human expression is both enabled and limited by brain size and organization.

  13. 13
    Cloud of Unknowing

    gpuccio,

    Regarding personal libraries, I have neither owned nor read The Cloud of Unknowing. I used to own The Meaning of Meaning, but I gave it and most of my other belongings to charity. Perhaps someone who has dispossessed himself of the meaning of meaning may lay claim to a cloud of unknowing.

    Puns aside, I am slightly familiar with semantics, and as a mystic I am highly focused on consciousness. But I honestly don’t see, even philosophically, the necessity of grounding meaning in consciousness. Putting on my scientist’s cap, I have to say that consciousness (in the sense of an entity’s perception of itself perceiving) is a hypothetical construct, and that I don’t know how to give it an operational definition. It seems to me that neuroscientists investigate neural correlates of “consciousness” only by downgrading the concept in such a way that any organism capable of forming a percept is conscious.

    Dembski defines specification in terms of entities capable merely of associating signs with events. He wisely avoids full-blown semantics and consciousness.

  14. Cloud of Unknowing:

    “as a mystic I am highly focused on consciousness”

    Me too.

    “But I honestly don’t see, even philosophically, the necessity of grounding meaning in consciousness”.

    I do see it. Meaning exists only in consciousness. It can be recognized only by consciousness. It is, as I suggested before, a feeling, although a cognitive one. For a more specific discussion about the definition of consciousness, see later.

    “Putting on my scientist’s cap, I have to say that consciousness (in the sense of an entity’s perception of itself perceiving) is a hypothetical construct, and that I don’t know how to give it an operational definition”

    In my personal language, that would be self-consciousness. More simply, in my language, consciousness means that there are events which are representations, and those events involve a subject (the perceiving “I”), and various objects (the modifications referred to the perceiving I as perceptions). In that sense, consciousness is present in any sensation, or higher representation of thought and feeling, and does not require “an entity’s perception of itself perceiving”, except in the fundamental form of the intuitive, innate perception that any “I” has of its own existence.

    Putting on my scientist’s cap, I would say that the basic definition of consciousness can and should be very simple and empirical: a phenomenon which exists, which is perceived directly by each one of us in himself, which is inferred by eac one of us in all other similar beings (on the basis of the evidence of that first, personal intuitive perception), and upon which any other knowledge of ourselves and of the world is based. That would be the definition. A phenomenological description of conscious events would ensue, with the basic description of a unit which refers to itself the modifications (the I) and the modifications themselves (the objects of perception). The point is, nothing of that is abstract or philosophical, it is only based on observation of existing events, although the basis of that observation is introspective.

    “Dembski defines specification in terms of entities capable merely of associating signs with events. He wisely avoids full-blown semantics and consciousness”

    That could be wisdom, or just prudence. The fact is, in my opinion, that specification is most simply defined as “any characteristic recognizable by consciousness as originating in consciousness”. That’s why specification is such an elusive concept. Pre-specification has its basis in the fact that some conscious observer has knowledge of a pattern before it is found in the “information unit” we are examining. Compressibility has its basis in the specific ability of conscious beings to recognize compressible patterns. Functional specification, the most important for us, is based on the ability of conscious beings to recognize purpose in a context. None of these concepts could exist outside of consciousness. All of them elude mere information theory. Dembski prefers to use the concept of “intelligent agent”, instead of “conscious agent”, but in reality intelligence cannot exist outside of consciousness, and is one of its manifestations.

    So, I believe that if we are to make any progress in the scientific frontiers we are exploring here (including origin of life and of biological information, and origin and meaning of consciousness and mind in biological beings) we do have to include semantics and discussion of the phenomenon of consciousness at a scientific level, completely abandoning the reductionist, materialistic approach.

  15. 15
    Cloud of Unknowing

    gpuccio,

    Does anybody really, really believe that random mutations and natural selection based on survival capacities have molded, step by step, the transition from chimps to humans in less than one million years, building the necessary brain structures which would allow man to intuit and understand the abstract principles of mathematics, and gradually discover the stunning fact that the world seems to work upon them?

    My working hypothesis, as a scientist, is that the combination of random variation and selection is efficacious because the world is extremely orderly. The very existence of an orderly universe is mysterious and marvelous to me. I see nothing profane in suggesting that order flowers, as it were, in an orderly universe. The big difference between most IDists and me is that they want evidence of interventions of a Designer with personal attributes, while I try to assume as little as possible about my Creator. I play at science, believing that inwardness, not outwardness (empiricism), is the way to make discoveries of ultimate importance.

    It is important to remember that while Newtonian, Einsteinian, and quantum mechanics are all incredible intellectual developments, they look nothing like one another. I see no historical basis for saying that mathematics is the language of nature, and that humans are imbued with the capacity to talk the language of nature. Math is the language of physical models, and we keep discovering that what we’ve said in the past about our observations of nature can be improved upon by saying something radically different.

    It’s worth mentioning something obvious that people forget: Our physical models are our inferences of the unobserved relations between our observations. We have never, and never will, essentially as a matter of definition, observe the “laws of nature.”

  16. Cloud of Unknowing:

    A few more comments.

    “My working hypothesis, as a scientist, is that the combination of random variation and selection is efficacious because the world is extremely orderly”

    Yes, but CSI is well different from order. I may agree with you that order in the universe is already evidence enough of a creator. But biological information is evidence in higher degree, and of different kind. For those who can doubt about the order in nature, biological information should be the final proof of a superior intelligence. As a mystic, you can neglect the difference between the “simpler” order in the universe and the “higher” complexity in biological beings. But as a scientist you can’t.

    “The very existence of an orderly universe is mysterious and marvelous to me”

    To me too.

    “I see nothing profane in suggesting that order flowers, as it were, in an orderly universe”

    I see nothing ascientific in suggesting that intelligent order flowers in a higher recognizable degree in living beings.

    “The big difference between most IDists and me is that they want evidence of interventions of a Designer with personal attributes, while I try to assume as little as possible about my Creator”

    Why do you say that? ID needs not assume anything about the Designer, not even that He is a Creator. ID needs only assume that the Designer is a conscious, intelligent being. Do you think your Creator is unconscious or unintelligent? (I know, you could say yes, and I could agree, in the sense that nothing limiting should be said of the Absolute, but you know I did not mean that!)

    “I play at science, believing that inwardness, not outwardness (empiricism), is the way to make discoveries of ultimate importance”

    Me too. But I don’t think that the two things are completely different. Inwardness is a very good basis for a correct empiricism.

    “It is important to remember that while Newtonian, Einsteinian, and quantum mechanics are all incredible intellectual developments, they look nothing like one another.”

    Yes, but they have two important things in common: they are all based on mathematics, and they all work in explaining much of the outer world. That’s enough, for me.

    “I see no historical basis for saying that mathematics is the language of nature, and that humans are imbued with the capacity to talk the language of nature”

    Agreed. My idea of nature is too rich to reduce its language to mathematics. But I was only making the point that nature, at least at some levels, seems to use mathematical structures, or at least to be cognizable by them. And there is no doubt that humans are able to understand and develop mathematical knowledge (not all of them, I know…)

    “Math is the language of physical models, and we keep discovering that what we’ve said in the past about our observations of nature can be improved upon by saying something radically different.”

    Absolutely true!

    “It’s worth mentioning something obvious that people forget: Our physical models are our inferences of the unobserved relations between our observations. We have never, and never will, essentially as a matter of definition, observe the “laws of nature.”

    I love you for saying that! I have spent hours on this blog desperately trying to reaffirm the epistemological difference between observed events (facts) and inferred models (theories), not only in response to the abuse in the darwinian field (you know, evolution is a fact and not a theory, and similar nonsense), but also with other IDers who seem to buy some of the lies of the darwinists. But you have said it perfectly right: a law is never observed, it is always inferred. And, I would add, no inference is ever final, least of all it can become a “fact” by winning the final prize of human dogmatism.

  17. 17
    Cloud of Unknowing

    gpuccio,

    You understand:

    (I know, you could say yes, and I could agree, in the sense that nothing limiting should be said of the Absolute, but you know I did not mean that!)

    I’m absurdly late in responding — I was tied up with a medical procedure, but…

    Inwardness is a very good basis for a correct empiricism.

    If so, you’ve made an excellent discovery I have not. I continue to seek, and who can say what I’ll live long enough to find?

    I sincerely respect your remarks, and I appreciate that you recognize and respond appropriately to the nuances of my comments.

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