The Unified Theory of the Nervous System
and Behavior
Cognitive Philosophy /Brain Theory by Steven Michael Harris
An early paragraph in the preface of this book states:
"But mathematics by itself does not and cannot empirically study human ideas; human cognition is simply not its subject matter. It is up to cognitive science and the neurosciences to do what mathematics itself cannot do - namely, apply the science of mind to human mathematical ideas. That is the purpose of this book."
This book might be looking somewhat in the right realm, but it is getting it all wrong. The fact is that current science does not yet know how to apply mathematics to the understanding of thought or cognition. But mathematical logic is the way to get the answers nonetheless. My premise is that all cognition is math. It is in the mathematical relationships in the nerve cells with each other and how the mathematical nature of nerve cell firings and connectivity accumulates into thought and other nervous system functions.
Where Mathematics Comes From takes on the idea that all of our mathematical knowledge is built up from very simple math abilities that are inherited and that it is always constructed from metaphor. They claim that metaphor is the basic unit in constructing the world of mathematics. But they offer no explanation of how such metaphor exists in thought. What constructs the metaphors? They don't take into account the possibility that the metaphor is a massive mathematical construct on its own and that it is far from being a basic unit or building block.
Millions of nerve cells are firing in a coordinated way when any metaphor is in mind. The factors that create an action potential in a nerve cell are mathematical. The basic unit is in the firing of those cells. The basic unit is the nature of the activity of a single cell in response to the signals coming from other cells.
Much of the confusion in the writing of this book is that the authors consider only the language of mathematics instead of considering the possibility of any non-language-based mathematics. They write as if the only mathematics that can exist is that which is communicated in the forms of language or symbols. This confusion is related to that confusion caused by those who define consciousness by the interior monologue (language in the brain) and it is easy to do this when it is only the language center of the brain that is able to communicate ideas in a book or a theory. (The language center of the brain can rarely comprehend anything other than language-based concepts. Concepts that can't be put into words are relegated to "emotional" understanding.)
It will probably take me a long time to communicate to the world the concept that language-based logic is actually a reduction of thought even though our language expands our world as a species. Mathematics that can calculate millions of dimensions of subject matter at the same time must be reduced to a form that can be articulated in a two-dimensional form (a variety of sounds following other sounds in relation to linear time) when applied to the creation of language. Any mathematics that cannot be reduced into such a form cannot be expressed in language. (Paradoxically a much more complex brain is required to handle this condensation of logic, to translate many-dimensional mathematics into a two-dimensional language form.)
Some of the authors conclusions are right, but not necessarily for the reasons they claim.
They list the aspects of mathematics that they consider romantic myths:
They write:
"But the more we have applied what we know about cognitive science to understand the cognitive structure of mathematics, the more it has become clear that this romance cannot be true. Human mathematics, the only kind of mathematics that human beings know, cannot be a subspecies of an abstract, transcendent mathematics. Instead, it appears that mathematics as we know it arises from the nature of our brains and our embodied experience. As a consequence, every part of the romance appears to be false, for reasons that we will be discussing."
They then list examples to support their theory that "most fundamental mathematical ideas are inherently metaphorical in nature:"
I'll need to comment on both lists. They are both right and wrong in various ways.
The mathematics that we know - the world of math that begins with learning to count as a child to the advanced theoretical mathematics explored in academia - is not disembodied just as the authors claim. It is biologically-based and it has been evolved from language and experience "metaphors" as it is a language of math (expressible in the symbols of verbal or written language). The only math most know is that which can be expressed through language and observed through human experience (any experience that itself is also expressible in language). The book repeats this point throughout as it goes through the evolution of math. But this should be obvious, as the world of acceptable math has evolved from the earliest, simplest concepts.
Later in the book there is examination of the region of the brain they claim that math is processed and this region is closely associated with kinesthetic, auditory, and visual sensory-motor systems. The same regions needed for language (spoken and written).
(There have been many studies that show how understanding can be received in non-language centers of the brain and understood, but without the ability to articulate such understanding when the language center of the brain was not getting the signal. Many of these studies have been in the realm of left-brain/right-brain studies, often using patients whose corpus callosum has been severed.)
The authors mention the possibility of a Platonic mathematics that exists without human cognition of mathematics. They mention that such a possibility is beyond our ability to perceive such as we can only perceive that that is within the powers of our biology. They are right about this... to a degree.
The mathematics that they describe as biologically-based math is really just a subset of biologically-based math: the biologically-based math that is language-based as well.
I've discovered the nature of non-language-based biological math and everything I write is a result of that discovery. Unfortunately it is very difficult to conceptualize such mathematics and just as difficult to articulate the nature of such because I'm required to use language to explain something that cannot be completely understood in the terms of language. (To conceive of such a mathematics is very difficult and requires some strange mind-control techniques as it is necessary to "dissociate" from language and emotional thinking in order to think in this way and then translate understanding that comes from this type of thinking back into language by going back into language-based thought. Most will never be able to do this completely. Autistic savants do this when they perform outstanding feats of mathematical calculation or artistic or musical performance.)
Read my earlier essay - Another very big clue - to get some sense of the limitations that have some effect on the nature of biological math (the fact that excitation flips to inhibition and leads to a gradual evolution of changes in sensitivity towards greater inhibition). Because of these limitations, the possibility of another, greater, Platonic math in the universe that is beyond the ability of our nervous systems to perceive or use in any way is an attractive possibility. (But only if you assume that mathematics can exist at all without perception of mathematics in some form: unless you insist on believing in a God. By the way, a belief in God will create conflicts in your thinking that will make an understanding of my theories almost impossible. All conflicts in thinking slow the brain. The brain speeds up when conflicts are reduced. Look at the limitations of logical thinking placed on tribes that are overwhelmed by many illogical superstitions.)
But the authors have not recognized the possibility that there is a much more complex and powerful non-language-based biological mathematics available to us as a realm in-between the mathematics they explain in the book and the theoretical Platonic math they discuss.
They dispute the idea that our mathematics is part of a universal, abstract, transcendent mathematics of the universe. Yes and no. Our mathematics is an aberrant form of the greater forms of mathematics that are possible. Our mathematics is severely limited by the mathematical limits in the way our language is expressed. We can only say one thing at a time so our math language can only say one thing at a time. Our non-language realms of mental processing can handle millions of different realms of decision-making (math) in unison and can bring together many different realms or dimensions of processing together into singular units of logic as well when this is possible or when useful patterns in such processing can be discovered.
Where Mathematics Comes From is a theory built entirely on an assumption of proof that there is an innate and simple mathematics built into our brains and into the brains of some other animals based on some flimsy "proof."
The arrogance of the authors' statements is significant. They put forth two questions:
Then they answer their own question:
"Question 1 is a scientific question, a question to be answered by cognitive science, the interdisciplinary science of the mind. As an empirical question about the human mind and brain, it cannot be studied purely within mathematics. And as a question for empirical science, it cannot be answered by an a priori philosophy or by mathematics itself. It requires an understanding of human cognitive processes and the human brain. Cognitive science matters to mathematics because only cognitive science can answer this question."
This statement demands a strong rebuttal. What a piece of crap!
In the current state of cognitive science the philosophers of consciousness can't agree on a definition for the word "consciousness." They don't know what cognition is and then they claim that they have the only approach for finding the answers about cognition.
Even though the authors are focusing on language-based-brain-and-mind-based mathematics when they talk about brain-and-mind-based mathematics, they do much better when they answer the second question:
Unfortunately they then start to corrupt this logic by referring to the incomprehensibility of Platonic math being akin to the incomprehensibility of God. Never trust "science" that brings religion in to support the argument. Such an approach never brings clarity.
In arguing their point they say that "it is only through cognitive science - the interdisciplinary study of mind, brain, and their relation - that we can answer the question: What is the nature of the only mathematics that human beings know or can know?" Wouldn't an "interdisciplinary" study of mathematical thinking include the discipline of mathematics itself as a contributor with possible insight? Especially when none of the fields of study has yet come up with an answer? Especially when the current mathematical approach to biology and behavior is so vague and inaccurate in spotting patterns?
They would be right when the say "that human mind-based mathematics uses conceptual metaphors as part of the mathematics itself" if they had the insight to know they were really describing language-based human mind-based mathematics.
Their next conclusion that "human mathematics cannot be a part of a transcendent Platonic mathematics, if such exists" is just not provable.
They don't consider the possibility that their observation that a "conceptual metaphor is a cognitive mechanism for allowing us to reason about one kind of thing as if it were another" exists because the mathematical properties of the non-language based thinking are consistent across the different realms of processing and perception.
The Brain's Innate Arithmetic?
The entire argument they present is based on the "proof" that there is a basic "hard-wired," genetically programmed ability to perform some simple math (language-based math) in some animals and in humans before any such ability is taught. Most of this proof is presented in Chapter 1 and the rest of the book is pretty much a synopsis of all mathematics, or the evolution of mathematics, constantly referring back to the proof presented in the first chapter. So the first chapter is where most of my disagreement will be based as well. It will not take that long.
They claim number discrimination by pre-language babies using some suspect logic.
They state as fact that "babies have the following numerical abilities":
The assumptions are that the abilities are of a mathematics (language-based mathematics) similar to what we use when we count in our heads "one, two, three..." but these assumptions come from understandable language-bias in interpreting observations from various studies. There are other ways to interpret the observations in these studies. I'll need to quote extensively from this chapter in order to present a fair argument.
"Slides were projected on a screen in front of babies sitting on their mother's lap. The time a baby spent looking at each slide before turning away was carefully monitored. When the bay started looking elsewhere, a new slide appeared on the screen. At first, the slides contained two large black dots. During the trials, the baby was shown the same numbers of dots, though separated horizontally by different distances. After a while, the baby would start looking at the slides for shorter and shorter periods of time. This is technically called habituation; nontechnically, the baby got bored."
"The slides were then changed without warning to three black dots. Immediately the baby started to stare longer, exhibiting what psychologists call a longer fixation time. The consistent difference of fixation times informs psychologists that the baby could tell the difference between two and three dots. The experiment was repeated with the three dots first, then the two dots. The results were the same. These experiments were first tried with babies between four and five months of age, but later it was shown that newborn babies at three or four days showed the same results (Antell & Keating, 1983). These findings have been replicated not just with dots but with slides showing objects of different shapes, sizes, and alignments (Strauss & Curtis, 1981). Such experiments suggest that the ability to distinguish small numbers is present in newborns, and thus that there is at least some innate numerical capacity."
They later state that the innate ability to count to three and sometimes to four is present at a very early age. But none of these studies has proved that counting (language-based math) has occurred. There are a variety of ways to approach this. It is more complicated to explain because innate biological math is immensely complicated in all animals with enough of a nervous system to be animated in complicated ways or to have any evolved sense of sight or hearing or to have a liver or...
One of the problems in this kind of logic is the assumption that the brain of any organism is completely the product of genetics, of a hard-wired pre-ordained ability like that of a machine. But a nervous system is an organization of logic that forms itself based on the mathematical principles it uses to operate and is affected by genetics but really operates as a mathematical logic that is constrained by the limits imposed by the genetics. Otherwise it would not be possible for a damaged brain to flexibly rearrange its logic in order to repair a function damaged by lesion or excessive cerebral fluid or whatever by using another region of the brain or another arrangement of the cells (following the logic of how brain cells interact with other cells).
Forget for a moment the fact that any animal that can intercept a moving object or that can calculate information based on arrangements of light affecting cells in one part of the body and calculate the necessary movements of thousands of muscle fibers in order to predict the existence of an object away from the body and reach out and find that object is using massive mathematical calculations to perform such feats (even though there is no language-based understanding of such math).
When the baby responds to a change from one to two, from two to three, rarely from three to four, but never from four to five - something besides "counting" is going on.
Think of it this way. The baby is reacting more significantly to observable change. The difference between one and two is a 100% change. The difference between two and three is a 50% change - still a significant change in amount or degree. Beyond three the amount of change is a minority of change. From three to four is a change of 33% and from four to five is a 25% change in amount, so the response to such change is less likely as there is a much smaller percentage of change. (If somebody gave you a glass with milk in it and added 25% to it when you were not looking, you might not notice the difference.) So this does not necessarily represent counting ability. (They never said if the study also tried to see a difference in response from three to five - a 66% change in amount.)
(Another way of saying the same argument: if you are napping and I increase the light in the room 100%, you are much more likely to respond by waking than if I just increased the light in the room by 25% or gradually increased the light by 100%.)
Another way of looking at it is that beyond three the brain might be using shorthand to assume further repetition so it does not have to continually process repeating objects. Remember that it might just take three observations of repetition to recognize a predictable pattern and after that an assumption of repetition might be the impulse. (The brain predicts objects - especially repeating objects such as the pattern in tile or wallpaper when filling in the blind-spot in vision, the reason that you can put a unique object into your blind spot and it disappears when repeating visual patterns surround the blind spot.)
You only need three points of observation on the arc of a ball to predict where it is going to land.
Any quantity beyond three might be inherently boring or just more than is needed when a pattern needs to be perceived.
In comedy, everything is setup in threes. The costume will always have three buttons, never four (unless when created by an amateur). The jokes are setup in threes. There are always three people walking into a bar. The anecdotes concerning the first two people setup the pattern. The third anecdote would confirm the pattern but in comedy there is always a switch in the pattern with the third anecdote which creates the humor. A joke that has four people walking into a bar would lose the audience if the punchline only came with the fourth anecdote (but would work if the switch occurred with the third person and then another switch was delivered concerning the fourth person).
It is our nature to assume that a pattern will go on indefinitely when established with three consistent examples of the pattern. (The ellipsis is three dots...) (Infinity is represented by three dots in mathematics...) This point is even mentioned by the authors in this book in a later chapter when talking about a different subject (making this essay of mine so much easier):
"Consider a sentence like John jumped and jumped again, and jumped again. Here we have an iteration of three jumps. But John jumped and jumped and jumped is usually interpreted not as three jumps but as an open-ended, indefinite number."...
"But verbs like swim, fly, and roll are imperfective, with no indicated endpoint. Consider sentences indicating iteration via the syntactic device of conjunction: John swam and swam and swam. The eagle flew and flew and flew. This sentence structure, which would normally indicate indefinite iteration with perfective verbs, here indicates a continuous process of swimming or flying. The same is true in the case of aspectual particles like on and over. For example, John said the sentence over indicates a single iteration of the sentence. But John said the sentence over and over and over indicates ongoing repetition. Similarly, The barrel rolled over and over indicates indefinitely continuous rolling, and The eagle flew on and on indicates indefinitely continuous flying. In these sentences, the language of iteration for perfectives (e.g., verb and verb and verb; over and over and over) is used with imperfectives to express something quite different - namely, an indefinitely continuous process."...
As a writer I would have chosen to assume I'd made my point earlier and moved on (because the joke should be over with the third example), but these authors (in the same way they repeated their points in other parts of the book) kept going on and on and on...
Back to the "proof" of inherent biological counting ability argued in the first chapter of Where Mathematics Comes From.
The psychologists use habituation (boredom) and then look for changes in response to decide that the baby can "count" from one to two or two to three and then ignore the possibility that the baby is bored by any change from four to five or beyond as the reason they don't respond and then these psychologist come to the conclusion that they have the ability to count to three or four and not beyond.
For many of the same reasons the logic of the experiments that "establish" an ability of babies to do simple arithmetic is suspect:
"Babies were tested using what, in the language of developmental psychology, is called the violation-of-expectation paradigm. The question asked was this: Would a baby at four and a half months expect, given the presence of one object, that the addition of one other object would result in the presence of two objects? In the experiment, one puppet is placed on a stage. The stage is then covered by a screen that pops up in front of it. Then the baby sees someone placing a second identical puppet behind the screen. Then the screen is lowered. If there are two puppets there, the baby shows no surprise; that is, it doesn't look at the stage any longer than otherwise. If there is only one puppet, the baby looks at the stage for a longer time. Presumably, the reason is that the baby expected two puppets, not one, to be there. Similarly, the baby stares longer at the stage if three puppets are there when the screen is lowered. The conclusion is that the baby can tell that one plus one is supposed to be two, not one or three.
"Similar experiments started with two puppets being placed on-stage, the screen popping up to cover them, and then one puppet being visibly removed from behind the screen. The screen was then lowered. If there was only one puppet there, the babies showed no surprise; that is, they didn't look at the screen for any longer time. But if there were still two puppets on the stage after one had apparently been removed, the babies stared at the stage for a longer time. They presumably knew that two minus one is supposed to leave one, and they were surprised when it left two. Similarly, babies at six months expected that two plus one would be three and that three minus one would be two. In order to show that this was not an expectation based merely on the location of the puppets, the same experiment was replicated with puppets moving on turntables, with the same results (Koechlin, Dehaene, & Mehler, 1997). These findings suggest that babies use mechanisms more abstract than object location.
"Finally, to show that this result had to do with abstract number and not particular objects, other experimenters had the puppets change to balls behind the screen. When two balls appeared instead of two puppets, four- and five-month-olds (unlike older infants) manifested no surprise, no additional staring at the stage. But when one ball or three balls appeared where two were expected, the babies did stare longer, indicating surprise (Simon, Hespos, & Rochat, 1995). The conclusion was that only number, not object identity, mattered.
"In sum, newborn babies have the ability to discern the number of discrete, separate arrays of objects in space and the number of sounds produced sequentially (up to three or four). And at about five months they can distinguish correct from incorrect addition and subtraction of objects in space, for very small numbers."
Many of my earlier arguments against the conclusions of the first listed experiments also apply to these experiments.
First, there are a couple of observations that I wish were mentioned here:
They declared that the babies expected two plus one to be three and three minus one to be two. They mentioned no experiment with three minus two. I'm assuming they would have tried it and perhaps there was not as much surprise. Does not matter much because it is still a change in volume of degree and may not have anything to do with counting as we understand it. The brain uses prediction to do everything and so would be predicting a visual experience. The changes in degree are the same in the first mentioned experiments so the same argument occurs here.
(There is an interesting implication that visual processing is math if the baby can discern the changes in number of objects before it can discern the much more difficult understanding of what the objects happen to be.)
Did they try adding one puppet to another and seeing how the baby would react if there were one big puppet behind the screen of the same volume that might be found with the volume of two puppets put together?
How many times did they display the right answer (one plus one = two) before they showed a different answer (one plus one = one)?
Did they try habituating the baby to a wrong answer (many times in a row showing one plus one = one) before showing a right answer and see what happens then?
Did the change from puppets to balls occur with relatively similar-sized objects or with objects of very different size?
All of these factors could change the results of the conclusions.
What if vision occurs in such a way that there is no difference between looking at one or looking at two, but we just think there is a difference? The center of our visual field could be very small so the illusion that we see two things might just be that we look at one thing here and then one thing there but the eye saccades are very quick so the two images are calculated to both be in the complete field of vision at the same time. The habit of putting such separate objects or views together into a whole is a form of non-language biological math so the memory of an object here and an object there would be natural for a moment. (How long can the screen stay up in the baby experiment before the baby forgets that another object is supposed to be in the field of view? Memory will have an effect on this experiment.) All visual understanding might be a mathematical event of a different sort.
My previous arguments also apply to adults in the authors' arguments about innate subitizing abilities:
"Subitizing
"All human beings, regardless of culture or education, can instantly tell at a glance whether there are one, two, or three objects before them. This ability is called subitizing, from the Latin word for "sudden." It is this ability that allows newborn babies to make the distinctions discussed above. We can subitize - that is, accurately and quickly discern the number of - up to about four objects. We cannot as quickly tell whether there are thirteen as opposed to fourteen objects, or even whether there are seven as opposed to eight. To do that takes extra time and extra cognitive operations - grouping the objects into smaller, subitizable groups and counting them. In addition to being able to subitize objects in arrays, we can subitize sequences. For example, given a sequence of knocks or beeps or flashes of light, we can accurately and quickly tell how many there are, up to five or six (Davis & Perusse, 1988). These results are well established in experimental studies of human perception, and have been for half a century (Kaufmann, Lord, Reese, & Volkmann, 1949). Kaufmann et al. observed that subitizing was a different process from counting or estimating. Today there is a fair amount of robust evidence suggesting that the ability to subitize is inborn. A survey of the range of subitizing experiments can be found in Mandler and Shebo (1982).
"The classic subitizing experiment involves reaction time and accuracy. A number of items are flashed before subjects for a fraction of a second and they have to report as fast as they can how many there are. As you vary the number of items presented (the independent variable), the reaction time (the dependent variable) is roughly about half a second (actually about 600 milliseconds) for arrays of three items. After that, with arrays of four or five items, the reaction time begins increasingly linearly with the number of items presented. Accuracy varies according to the same pattern: For arrays of three or four items, there are virtually no errors. Starting with four items, the error rate rises linearly with the number of items presented. These results hold when the objects presented are in different spatial locations. When they overlap spatially, as with concentric circles, the results no longer hold (Trick & Pylyshyn, 1993, 1994)."
There is a lot wrong with these statements about subitizing.
First, it is wrong to assume that adult subitizing is the same thing that is happening that allows newborns to make distinctions between small numbers. First of all, the adult has developed a language-based form of thinking and the baby has no language. The baby is in the early stages of developing all functioning and has no ability to articulate what is happening to researchers so that they can be so sure of their conclusions. The reason a baby is more or less interested in an event can only be guessed at. Baby logic is probably very different from researcher logic.
What if all perception is a massively complicated "subitizing" but not in the form of language? The adult tested for subitizing ability is being tested with the language center giving all the answers. So the limits of subitizing are being tested with the language center using language center logic to find the answer (very fast counting).
And if adult subitizing in this way is supposed to be the same thing as the subitizing of babies who react slightly differently to a change in the number of objects, wouldn't the appropriate experiment to give adults not be one where they count (express mathematics in language) but notice change in a non-verbal way? For instance, a non-verbal change in reaction to a picture that keeps flashing over and over until there is a different picture that has 99 instead of 100 objects. Or 22 instead of 23, to make it easier. Reaction times would be much faster in the adults to such change than in the experiment where they give a verbal counting of the number of items in a group. (Keep in mind that many or most adults will be unable to avoid thinking in the terms of language after a lifetime of the habit.)
A language-based response in an adult is a very different event than a non-language-based response in an infant.
The authors conveniently ignore the existence of autistic savants who have the ability to instantly subitize very large groupings of objects. (One of the clues to understanding how they have this ability is to remember that those with autism are often shut off from language and emotion and could have the ability to shut language and emotion on and off very quickly. They make a particular kind of face, show stress in certain muscles, when they go into the state they seek when using the unusual abilities and this probably represents a shutting-off of the language and emotional centers in a form of meditative control of focus they have the ability to achieve. When they do this instantaneous subitizing or calculations of other sorts, they quickly go into this state and then pop out of it to express the answer in language.
By the way, objects lined up in a row, as opposed to concentric circles, are easier to subitize in the mentioned experiments because objects lined in a row are ordered much in the same way that words are lined up in a linear fashion and language thinking works better with spatial arrangements similar to language as opposed to concentric circles that are spatially arranged in a way that is very different from the way that language is arranged on a page.
Later the authors write:
"The Numerical Abilities of Animals
"Animals have numerical abilities - not just primates but also raccoons, rats, and even parrots and pigeons. They can subitize, estimate numbers, and do the simplest addition and subtraction, just as four-and-a-half-month-old babies can. How do we know? Since we can't ask animals directly, indirect evidence must be gathered experimentally.
"Experimental methods designed to explore these questions have been conceived for more than four decades (Mechner, 1958). Here is a demonstration for showing that rats can learn to perform an activity a given number of times. The task involves learning to estimate the number of times required. The rats are first deprived of food for a while. Then they are placed in a cage with two levers, which we will call A and B. Lever B will deliver food, but only after lever A has been pressed a certain fixed number of times - say, four. If the rat presses A the wrong number of times or not at all and then presses B, it is punished. The results show that rats learn to press A about the right number of times. If the number of times is eight, the rats learn to press a number close to that - say, seven to nine times.
"To show that the relevant parameter is number and not just duration of time, experimenters conceived further manipulations: They varied the degree of food deprivation. As a result, some of the rats were very hungry and pressed the lever much faster, in order to get food quickly. But despite this, they still learned to press the lever close to the right number of times (Mechner & Guevrekian, 1962). In other series of experiments, scientists showed that rats have an ability to learn and generalize when dealing with numbers or with duration of time (Church & Meck, 1984).
"Rats show even more sophisticated abilities, extending across different action and sensory modalities. They can learn to estimate numbers in association not just with motor actions, like pressing a bar, but also with the perception of tones or light flashes. This shows that the numerical estimation capacity of rats is not limited to a specific sensory modality: It applies to number independent of modality. Indeed, modalities can be combined: Following the presentation of a sequence of, say, two tones synchronized with two light flashes (for a total of four events), the rats will systematically respond to four (Church & Meck, 1984)."
The authors then go on to explain how some adult monkeys have been able to perform mathematical functions that go a little beyond what they claim the babies are doing in the earlier mentioned experiments.
This is pretty much the entire argument of the authors about the existence of a simple inherited mind-and-brain-based mathematics that is the basis for all math. Then they give a big math lesson that builds from the ability to count to three or four and to add or subtract one or two from these small numbers. (All math, language-based math I say, comes from this as they rightly claim.)
The simple inherited math they claim is the extent of our innate mathematical ability. But it is a form of mathematical logic that is language-based.
Yes, perhaps there is no math without any perception by a life form that creates that math. But the only math they claim is the language-based math.
If there were no life forms on earth, events would occur according to mathematical principals of some kind. Gravity and chemistry and magnetism and heat, etc. would cause things to occur according to mathematical principals when micro and macro objects come into contact with one another and react according to mathematical principles. A Platonic math might make much greater sense of such events without any organisms to try to assume the understanding that is possible to a life form.
But when a life form takes in the evidence of patterns of light into the retina, and then assembles the mathematical evidence of patterns in that light to predict that an object (not yet encountered) is on the way and then reacts to catch a ball, for instance, the organism is not reacting to the object but to the prediction of the object based on evidence that would disappear with just the closing of the eyes. Massive mathematical calculations are occurring even if we don't have a language-based method of explaining such math. It is still math. But a different kind that we have not figured out yet. It is not magic, it is not following the directions of a higher-power (unless you realize that the perception of a higher power is just the inexpressible understanding that there is intelligence within us that can't be explained by our language center), it is not the hard-wired response of a machine-like structure... it is math. Emotional math.
A bird that navigates several thousand miles to a particular piece of land has no ability to explain how this is done, has no language of math, but is using math just the same to find that spot. Navigation by stars, or landmarks, or by the earth's magnetic field all require mathematical principals in order to succeed. Enormous mathematical calculations are required just to hold our bodies erect. All of the mathematical and computer knowledge in the world is still not enough for scientists to be able to achieve a simulation of some of our simplest abilities. It is math in our bodies even if it is not a language-based (expressible in words) ability at this time.
I think I've said enough to throw doubt into the concepts the authors of Where Mathematics Comes From are stating as proofs of their theory in the first chapter of their book. The rest of the book is based on that first chapter so the rest of the book does not hold water either. They do make a good argument for the evolution of math from simple language-based concepts as they are discussing the language of math.
I can say that the book has a very nice cover.
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Many of the problems of medicine, biology, psychology and philosophy require an understanding of the basic mathematical principles behind how the nervous system does what it does to achieve function and experience, and that mathematics is not explained using narrowly-focused statistics. Understanding how this math works will be the tool for the discovery of many answers of great importance to humanity. The case for this concept and the offering of an explanation of this kind of math is made in the many essays of this website.
On these pages you will find ideas that should haunt you. Included are new concepts in science, medicine, sociology, evolutionary psychology, philosophy and more...
This website and the podcasts of Everyone's Revolution explain how the brain creates the mind, but many side issues must be resolved in order to teach this material. Once you realize that the "hard problems" are really the first problems to be answered, you then have a tool for changing all of science and medicine by explaining a massive number of discoveries that will fall into line in order to unify the evidence. All of the evidence is good. The interpretations of the evidence are mistaken in many cases. For ten years now there have been new discoveries of evidence that all move in the direction of supporting this theory (or this school of many theories) and its predictions. Quite a few people have started to pay attention to this theory as well.
