Drawing parallels between the contrasting Dehaene vs. Tononi theories and Einstein’s theory of general relativity vs. Newton’s gravitational theory (Reardon, 2019) is, I’m afraid, disrespectful of not only the gravity of the theories of Newton and Einstein, but also of the very notion of theory.
First, what is theory? It is clearly not speculations spiced with wishful thinking (of eminent scientists; cf. Vazire, 2017). A theory of a thing is a part of the thing that accounts for [almost] everything about the thing. A theory is an abstract general—a definite thing—and certainly not, as Koch claims, “very flexible” (Reardon, 2019) conforming to the fancies of famous scientists.
Next, how do we theorize? According to Professor F. William Lawvere, “any real object has an infinite complexity of internal structure; fortunately we are usually able through experiment and study to find out the most important structure of an object, in the sense that the most important structure of the object influences or largely determines all the other structures of the object and that it is mainly responsible for the workings of the object. Thus the mathematical method consists of taking the main structure by itself as a first approximation to a theory of the object, i.e. mentally operating as though all further structure of the object simply did not exist” (Lawvere, 1972, pp. 9-10). Furthermore, “regarding the scientific process of idealization, there are two equally important aspects. The idealization itself often consists of assuming that, of the many forces acting in a situation, one ‘main’ force is the only force. In Galileo’s investigation of falling bodies, this one force is gravity. Such idealization can lead to very far-reaching development of theory; in the example of gravity, it led Galileo, Newton, Jacobi, Hamilton, Einstein, and others to theory which is constantly used in terrestrial and celestial navigation. But the second equally important aspect of the scientific process of idealization is this: in applying the developed theory to new situations, one must constantly remain conscious of the likelihood that forces other than the ‘main’ idealized ones are also acting and sometimes becoming ‘main’ forces themselves. Galileo knew quite well that if instead of a cannon ball or ball of wood, a dry leaf were dropped from Pisa’s tower, friction and wind would be significant forces determining its fall; one might even observe the ‘paradox’ that sometimes the leaf falls upward, which does not mean that the theory of pure gravity would be wrong, but rather that a more comprehensive pure theory would be better applicable to the case” (Lawvere and Schanuel, 2009, pp. 308-309).
There is another equally important aspect to the development of scientific theories. Given a universe of discourse, we need to identify its basic notion, and then develop the mathematics of the basic notion. In the case of Newtonian physics, ‘change’ is the basic notion. Upon recognizing ‘change’ as the basic notion, the mathematics of change—calculus—was developed, which formed the basis for the development of physical theories (Lawvere, 1999, p. 412). Just as in the case of physics, we need a mathematical framework of the basic notion of consciousness studies in order to develop a scientific theory of consciousness. What is the basic notion of consciousness studies? Representation is the basic notion of the study of consciousness (Chalmers, 2006; Núñez et al., 2019, p. 782). Given that representation is the basic notion of consciousness, what is the mathematics of representation? Functorial semantics is the mathematics of representation (Lawvere, 1963, 2004). Briefly, given a universe of discourse (or a category of particulars), a representation or model of a particular is an interpretation of the theory of the given category abstracted from measured properties of the particulars (Lawvere, 1994, pp. 44-47). As such, a scientific theory of consciousness can be developed based on the mathematical framework of functorial semantics (Posina, Ghista, and Roy, 2017). Equally importantly, we also have the mathematics of reconstructing objects from their representations (Lawvere and Rosebrugh, 2003, pp. 125-126, 148-152), which takes on added significance in view of our working definition of perception: “the action of the mind by which it refers its sensations to an external object as their cause” (Albright, 2015, p. 22). No less important is the fact that we have mathematical definitions of the notions of unity (or cohesion) and quality (Lawvere, 2007), which figure prominently in the study of conscious experiences.
Summing up, it doesn’t seem like a good idea to draw parallels to Newton (cf. Ramachandran and Hirstein, 1997) until one has acquired a good grasp of the depth of the contributions of Newton to physics (e.g. Chandrasekhar, 2003). More importantly, given that we have the mathematics of representation—functorial semantics—a scientific theory of consciousness can be developed by applying functorial semantics to the category (or the totality) of conscious experiences (Koch, 2018, p. S9).
References
Albright, T. D. (2015) Perceiving. Daedalus 144: 22-41.
Chalmers, D. J. (2006) The representational character of experience. In B. Leiter (Ed.), The Future for Philosophy (pp. 153-181) New York: Oxford University Press.
Chandrasekhar, S. (2003) Newton’s Principia for the Common Reader. New York: Oxford University Press.
Koch, C. (2018) What is consciousness? Nature 557: S9-S12.
Lawvere, F. W. (1963) Functorial semantics of algebraic theories. Proc Natl Acad Sci USA 50: 869-872.
Lawvere, F. W. (1972) Perugia Notes: Theory of Categories over a Base Topos. Perugia: Universita’ di Perugia.
Lawvere, F. W. (1994) Tools for the advancement of objective logic: Closed categories and toposes. In J. Macnamara and G. E. Reyes (Eds.), The Logical Foundations of Cognition (pp. 43-56) New York: Oxford University Press.
Lawvere, F. W. (1999) Kinship and mathematical categories. In P. Bloom, R. Jackendoff, and K. Wynn (Eds.), Language, Logic, and Concepts (pp. 411-425) Cambridge, MA: MIT Press.
Lawvere, F. W. (2004) Functorial semantics of algebraic theories and some algebraic problems in the context of functorial semantics of algebraic theories. Reprints in Theory and Applications of Categories 5: 1-121.
Lawvere, F. W. (2007) Axiomatic cohesion. Theory and Applications of Categories 19: 41-49.
Lawvere, F. W. and Rosebrugh, R. (2003) Sets for Mathematics. Cambridge, UK: Cambridge University Press.
Lawvere, F. W. and Schanuel, S. H. (2009) Conceptual Mathematics: A First Introduction to Categories. Cambridge, UK: Cambridge University Press.
Núñez, R., Allen, M., Gao, R., Rigoli, C.M., Relaford-Doyle, J., and Semenuks, A. (2019) What happened to cognitive science? Nat. Hum. Behav. 3: 782-791.
Posina, V. R., Ghista, D. N., and Roy, S. (2017) Functorial semantics for the advancement of the science of cognition. Mind & Matter 15: 161-184.
Ramachandran, V. S. and Hirstein, W. (1997) Three laws of qualia: What neurology tells us about the biological functions of consciousness. Journal of Consciousness Studies 4: 429-457.
Reardon, S. (2019) Rival theories face off over brain’s source of consciousness. Nature 366: 293.
Vazire, S. (2017) Our obsession with eminence warps research. Nature 547: 7.
The characterization of artificial intelligence (AI) as alchemy, albeit emotive, is merely an acknowledgment of the lack of understanding of the workings of the underlying algorithms (1, see also 2). In developing an explicit theoretical understanding of intelligence, following in the footsteps of physics makes perfect sense. However, the toy problem (black-and-white images instead of color photos) prescribed for gaining insights into recognition algorithms appears to be diagnostic of a deeper problem with contemporary AI, which is getting lost in the here & now of immediate solutions and losing sight of the big picture: development of the science of intelligence (3). Here we suggest a different approach: mathematical knowing as a Bohr atom of knowing, which can serve as a solid foundation to build the science of human intelligence (subsuming reasoning, perception, and cognition, among others).
But how, and why?
First, science is a hallmark of human intelligence. As such, a scientific account of science constitutes the science of intelligence. Although we do not have the science of the development of scientific theories and models, we have functorial semantics: a mathematical account of abstracting theories and building models of various categories of mathematical objects (4, 5). Taking a cue from Galileo’s investigation of a simple motion—motion of falling bodies—that served as a foundation for the science of motion, we suggest investigating the functorial semantics of mathematical knowing as an elementary form of the more elaborate scientific knowing in particular and ordinary cognition in general (6). Functorial semantics spells out the mathematics of going from [categories of] mathematical objects to their measured properties, abstract theories, and concrete models; and this functorial calculus can provide an overarching mathematical framework needed for the advancement of AI, somewhat analogous to way calculus provided the mathematical framework needed for the advancement of physics (7).
References
1. M. Hutson, Science 360, 478 (2018).
2. M. Buchanan, Nature Physics 14, 326 (2018).
3. S.J. Russell, P. Norvig, Artificial Intelligence: A Modern Approach (Pearson Education Limited, 2016), pp. 1-31.
4. F.W. Lawvere, Reprints in Theory and Applications of Categories 5, 1 (2004).
5. F.W. Lawvere, in The Logical Foundations of Cognition, J. Macnamara, G.E. Reyes, Eds. (Oxford University Press, 1994), pp. 43-56.
6. V.R. Posina, D.N. Ghista, S. Roy, Mind & Matter 15, 161 (2017).
7. F.W. Lawvere, in Language, Logic, and Concepts, P. Bloom, R. Jackendoff, K. Wynn, Eds. (MIT Press, 1999), pp. 411-425.
What is this thing–BRAIN–that is the root cause of excruciating brain aches of many working neuroscientists? Brain is what brain does! In this respect, Hippocrates nailed it: “from the brain, and from the brain only, arise our pleasures, joys, laughter and jests, as well as our sorrows, pains, griefs and tears. Through it, in particular, we think, see, hear, and distinguish the ugly from the beautiful, the bad from the good, the pleasant from the unpleasant. It is the same thing which makes us mad or delirious, inspires us with dread and fear, whether by night or by day, brings sleeplessness, inopportune mistakes, aimless anxieties, absent-mindedness, and acts that are contrary to habit. These things that we suffer all come from the brain” [1].
But how? There have been many ideas of the brain: hydraulic pump, telegraph, and computer, just to name a few [2]. In light of the fact that we now find all these conceptualizations of the brain unsatisfactory, let us consider an all too obvious idea. To begin with, there are stimuli and then there is the brain. When presented with a stimulus, some neurons in the brain fire action potentials, while others rest. (Of course, there is more to the story, which we will get to in a couple of paragraphs.) The neural response to stimuli can be construed as a brain-valued property of the stimulus, i.e., p: S –> B, where p is a property of the stimulus S and the brain B is a property type [3]. A more familiar property type is the set of natural numbers N = {0, 1, 2…}, wherein lie the values of properties such as the population of a country. The set of real numbers is another property type, where properties such as length and weight are valued.
What next? With the brain as a property type, we find that the stimulus can be reconstructed from the brain-valued properties of the totality of brain-valued properties of the stimulus. Simply put, an object can be recovered from the properties of the properties of the object [4]. With ease of exposition in mind, we consider an [extreme] abstraction of the brain: brain B is a two-element set B = {not firing, firing}, which are the two states neurons in the brain can be in. In this case, we find that there can be perceived elements with no underlying actual elements, i.e. the property type brain B = {not firing, firing} leaves room for illusions. However, a minor refinement of our abstraction of the brain, which incorporates the fact that the firing rate of a neuron can remain constant or increase or decrease in response to a stimulus, i.e. with the property type brain as a three-element set B = {decreased firing rate, constant firing rate, increased firing rate}, we find that we get illusion-free perception [5, 6, see also p. 180 in ref. 3]. Treating the brain as a two- or three-element set is, admittedly, a crude approximation. Surely, the brain, when considered as a property type, is a much more structured (cohesive-and-variable) category than the structureless (discrete-and-constant) category of sets. But, our main objective here is to introduce the idea: brain is a property type, which admits considerable refinement and development. I hope the idea–brain is a property type–will make it into the future editions of the book “The Idea of the Brain” [7].
Well, what about mind, consciousness, cognition, and the self that got many of us into neuroscience in the first place? The idea–brain is a property type–is based on a fragment [see refs. 3, 4] of a comprehensive mathematical understanding of mathematics within the framework of F. William Lawvere’s functorial semantics [8] and objective logic [9], which have been brought to bear on cognitive neuroscience [10, 11].
Originally published: https://www.science.org/doi/10.1126/science.abc0421
References
1. Kandel, E. R., Schwartz, J. H., Jessell, T. M., Siegelbaum, S. A., and Hudspeth, A. J. (2013) Principles of Neural Science, New York: McGraw Hill Education, p. iv.
2. Gomez-Marin, A. (2020) A history of the metaphorical brain. Science 368, 375.
3. Lawvere, F. W. and Schanuel, S. H. (2009) Conceptual Mathematics: A First Introduction to Categories, (2nd ed.) Cambridge, UK: Cambridge University Press, pp. 81-85, 370-371.
4. Lawvere, F. W. and Rosebrugh, R. (2003) Sets for Mathematics, Cambridge, UK: Cambridge University Press, pp. 125-126, 148-152.
5. Lawvere, F. W. (2004) Functorial concepts of complexity for finite automata. Theory and Applications of Categories 13, 164-168.
6. Posina, V. R. and Roy, S. (2020) Objective logic of consciousness. 14th Nalanda Dialogue, Nava Nalanda Mahavihara, Nalanda.
7. Cobb, M. (2020) The Idea of the Brain: The Past and Future of Neuroscience, New York, NY: Basic Books.
8. Lawvere, F. W. (2004) Functorial semantics of algebraic theories and some algebraic problems in the context of functorial semantics of algebraic theories. Reprints in Theory and Applications of Categories 5, 1-121.
9. Lawvere, F. W. (1994) Tools for the advancement of objective logic: Closed categories and toposes. In The Logical Foundations of Cognition, J. Macnamara and G. E. Reyes, eds., New York, NY: Oxford University Press, pp. 43-56.
10. Posina, V. R., Ghista, D. N., and Roy, S. (2017) Functorial semantics for the advancement of the science of cognition. Mind & Matter 15(2), 161-184.
11. Posina, V. R. (2020) Hard, harder, and the hardest problem: The society of cognitive selves. Tattva – Journal of Philosophy 12(1), 75-92.
To guide the learning, development, and use of mathematical sciences, the laws of possible rational passage from one assertion to another need to be made explicit.
But assertions are phrased in terms of concepts, therefore it is also urgent to make explicit the laws of possible rational passage from one concept to another.
The theory of categories, developed by Eilenberg, Mac Lane, Kan, and Grothendieck is the most extensive approximation to an understanding of the laws of passage from one concept to another.
Individuals do not set the course of events; it is the social force.
The singular purpose of education is to nurture the universal yearning for understanding. Here we show how a nurturing pedagogy naturally resolves the “learning crisis”, with the solution manifesting as human development (alluded to in Muralidharan and Singh, 2021). In doing so, we spell out–for further discussion–how the historic Indian education reforms embodied in the National Education Policy 2020 (NEP) can be implemented. Furthermore, the nurturing pedagogy that we advocate invariably results in the culture of research excellence that the national education policy calls for (NEP, pp. 45-46).
First, what is understanding? Understanding is organized knowledge, wherein bits and pieces of knowledge are organized into a cohesive body of understanding “so that the new ideas and methods collected and developed as one goes through life can find their appropriate places” (Lawvere and Schanuel, 1997, p. xiii). As such, our education reforms calls for placing: “emphasis on conceptual understanding rather than rote learning” (NEP, p. 5).
Aren’t concepts abstract and therefore difficult to understand? Yes, concepts and ideas are abstract. But, “we all begin gathering mathematical ideas in early childhood, when we discover that our two hands match, and later when we learn that other children also have grandmothers, so that this is an abstract relationship that a child might bear to an older person, and then that ‘uncle’ and ‘cousin’ are of this type also” (Lawvere and Schanuel, 1997, p. xiii). This natural fluency in abstract thinking turns into fear of mathematics as a result of the teaching-to-test education.
We (> 50 years old) all remember memorizing multiplication tables all the way up to 16 x 16, not to mention being academically rewarded for the speed with which we recited–without pausing–the multiplication tables correctly in the class. With technological advances (e.g. calculators), this knowing by heart is of little value. A more fruitful exercise–with lasting value–would have been asking: Why 2 x 3 = 6? Yes, it is a sensible question (ibid, p. 9). A first step in answering this question is to state explicitly what we did in answering 2 x 3 = ? We picked a number (6) from all the numbers that there are. In order to pick one out of many, the picked one must be distinct from the rest, i.e. it must be unique. Further reflection reveals that that uniqueness is with respect to the given operation (multiplication) and the given factors (2 and 3), which simply means that if we change the operation to addition, i.e. 2 + 3 = , then the number on the right-hand side would be different (5); so is the case with changing the factors, i.e. 3 x 4 is no longer 6. The key point to note here is the uniqueness of sums and products. Working slowly along these lines leads to the definitions of PRODUCT and SUM, which capture the mathematical content of everyday concepts ‘AND’ and ‘OR’, respectively (ibid, pp. 216-222).
In searching for basic notions and in constructing fundamental concepts, mathematicians think slowly. In contrast, pre-university college students are encouraged to calculate, for example, the number of subsets (or parts) of a set X–at lightning speed–using the formula 2^|X|, but rarely nudged to pause and ponder: Where did the 2 in the formula 2^|X| come from? 2 is the two-element set {false, true}. This relationship between sets and logic is not visible in the current mathematics curriculum, where set theory and logic are taught as separate subjects. Conceptual Mathematics transcends these artificial boundaries separating algebra, arithmetic, calculus, geometry, and logic, thereby bringing the unity of mathematics into clear focus for all to see and use (ibid, p. xiii). In fact, “explicit use of the unity and cohesiveness of mathematics sparks the many particular processes whereby ignorance becomes knowledge” (Lawvere, 1991, p. 2; see also Ehresmann, 1966).
What exactly is the unity of mathematics? For example, logic is the algebra of parts (Lawvere and Rosebrugh, 2003, pp. 193-212; Lawvere and Schanuel, 1997, pp. 335-352; see also Logic of Mind). Algebra is the opposite of geometry (Lawvere and Schanuel, 2009, pp. 81-85, 369-371; Lawvere, 2016). In everyday terminology, if we think of geometry as looking at [the back of] a car in front of us, algebra is like looking into the rearview mirror to see [the front of] a car behind us. Notice that we can overtake the car in front of us to see its front in the rearview mirror just as we can let the car behind us overtake us to see its back, which is an informal description of how we solve difficult geometric problems by translating them into algebra (and vice versa). Next, how does the unity–transcending artificial disciplinary boundaries–of mathematics contribute to the advancement of mathematics? Upon recognizing the unity of mathematics, Professor F. William Lawvere showed that the foundation of mathematics is within mathematics, i.e. mathematical foundation does not need a language distinct from that of mathematics (Lawvere and Rosebrugh, 2003, pp. ix-x, 235-236; Lawvere, 2003). In this spirit, our new education policy emphasizes multidisciplinarity and unity of knowledge (NEP, p. 5).
Taking cognizance of the fact that “young children learn and grasp nontrivial concepts more quickly in their mother tongue” (NEP, p. 13), our national education policy calls for teaching mathematics in students’ mother tongue. The medium of instruction in government high schools is, in fact, students’ mother tongue, but the medium of instruction of all higher education (beginning with undergraduate engineering) is English. As a result of which, even bright students, who graduate from government high schools with excellent grades, all of whom are from low-income families, find it difficult to excel in higher education. In an effort to remove this additional burden placed on the already [financially] disadvantaged student population, our national education policy calls for setting up of Indian Institute of Translation and Interpretation (IITI) dedicated to translating advanced scientific texts into students’ mother tongues, so that all students, irrespective of their family financial status, can excel in higher education (NEP, pp. 53-55).
In implementing any policy, especially one of vast reach and immense impact such as the Indian national education policy, the first step is more than a step: it is the foundation on which the future of India rests; it is the rocket that launches our national education (policy satellite) into an orbit of research excellence! In light of the gravity of the first step in implementing our national education policy, we find that Bengaluru is well placed to house IITI. Why Bengaluru? Bengaluru is reflective of the diversity of India. Kannadigaru, with their gracious hospitality, transformed Bengaluru into a home of all Indian languages. With people speaking every Indian language living and making a living in Bengaluru, Bengaluru has all the resources needed for fulfilling the mandate of IITI.
Another equally immediate question that the IITI needs to address is: Which advanced scientific text to translate? We suggest translating Conceptual Mathematics (Lawvere and Schanuel, 2009) into Kannada as ಪರಿಕಲ್ಪನಾ ಅಧ್ಯಯನ (Parikalpane Adhyayana, based on the meaning of the etymological roots of the word: mathematics), and making the Kannada textbook ಪರಿಕಲ್ಪನಾ ಅಧ್ಯಯನ (Parikalpane Adhyayana) freely available online and in print to students attending government schools. Why Conceptual Mathematics or ಪರಿಕಲ್ಪನಾ ಅಧ್ಯಯನ (Parikalpane Adhyayana)? Before we answer this question, we must note that Lawvere and Schanuel (2009) Conceptual Mathematics textbook is a first introduction to categories (or universes of discourse). A universe of discourse, say, a category of cats is one in which every cat in the category has the essence (in the sense of the way parts of a whole stick together; see Lawvere and Schanuel, 2009, p. 146; Lawvere, 2015) ‘catness’; and every transformation of any cat is a natural transformation (e.g. transformation of a small cat into a big cat, preserves the catness that characterizes the category of cats). Therefore, what we refer to as Conceptual Mathematics is also known as Category Theory or Theory of Naturality (Eilenberg and MacLane, 1945; see also Lawvere, 2006, p. 2).
Now we address–from a broad perspective–the question:
Why Conceptual Mathematics or ಪರಿಕಲ್ಪನಾ ಅಧ್ಯಯನ (Parikalpane Adhyayana)?
There is one thing that we have been doing since the dawn of free will, which is socializing. We have been telling stories and we have been listening to songs. In participating in the conscious practice of communicating with one another, we abstracted the essence–grammar–of language, which guides the practice of expressing our thoughts in words and sentences, such that they are understood in the intended sense. Recognizing the importance of understanding one another in sustaining social unity, we teach grammar early in education.
In interacting with our fellow human beings, we are also in constant interaction with reality. This interaction, which includes telling and listening, takes the form of thinking about things and making things we think of (for example, pots, music, and friendships). Participation in the conscious practice of thinking-and-making led to the abstraction of its essence–science in general and mathematics in particular–that we can use to live in harmony with nature. Conceptual Mathematics or ಪರಿಕಲ್ಪನಾ ಅಧ್ಯಯನ (Parikalpane Adhyayana) is all about this grammar of life: the essence of thinking about things and making things we think of (p. 6; see also Lawvere, 1994a, 2003). Just as we teach grammar–the essence of telling and listening–so that we can communicate and collaborate to sustain our society, we also need to teach Conceptual Mathematics or ಪರಿಕಲ್ಪನಾ ಅಧ್ಯಯನ (Parikalpane Adhyayana)–the essence of thinking and making–so that we can continue to search for evermore proper answers to the most basic question: How should we live our lives in harmony with reality?
Now, we present a few examples–all too familiar in our everyday experience–of what we learn in Conceptual Mathematics or ಪರಿಕಲ್ಪನಾ ಅಧ್ಯಯನ (Parikalpane Adhyayana; see also Science of Knowing).
The way we think about things depends on the nature of the things. For example, I cannot be both dead and alive, but India’s boundary is both India and not India (Lawvere, 1994a, p. 48; Lawvere, 2003, pp. 214-215; Lawvere and Rosebrugh, 2003, p. 201).
All that there is to know about any object is in its relationships (Lawvere and Schanuel, 2009, pp. 369-371). For example, you can find all about me by looking at my contacts.
A part of a whole is both itself and its relationship to the whole (Lawvere, 1994a, p. 53). For example, my head is not only my head but also includes its connections to my body.
Being determines Knowing (Lawvere, 2003, p. 217). For example, I cannot find the name of a child, who does not understand English, by asking: What is your name?
What is it good for? Imagine a child pointing to a pen and asking: what is it? We describe PEN in terms of ‘what it is good for’, which is WRITING. This is also how we define mathematical objects (Lawvere and Rosebrugh, 2003, pp. 26-29; Lawvere and Schanuel, 2009, p. 334).
Equipped with a conceptual understanding, which develops slowly, of these examples and their vast scientific reach (Lawvere, 1992), students of Conceptual Mathematics or ಪರಿಕಲ್ಪನಾ ಅಧ್ಯಯನ (Parikalpane Adhyayana) can extract the mathematical (meaning: teachable and learnable; Lawvere, 1999) content of any subject matter. (This is not unlike the scenario, wherein learning grammar is useful in writing epics, executive summaries, etc.) As such, teaching Conceptual Mathematics or ಪರಿಕಲ್ಪನಾ ಅಧ್ಯಯನ (Parikalpane Adhyayana) to pre-university college, undergraduate, post-graduate, and doctoral students will endow them with the intellectual freedom needed to formulate and answer original scientific questions.
How does this nurturing pedagogy translate into research excellence? It is in teaching–teaching in an understandable manner–that we see the world anew: “the need to explain daily for students is often the source of new mathematical discoveries” (p. 13; see also Lawvere, 2003, 2005a). One such discovery is the category of sets (Lawvere, 1994b, 2005b). Along these lines, in explaining that “the motion of a rock means more than its track” (Lawvere and Schanuel, 2009, p. 3), we find that that “more” is the space containing the track (ignoring this obvious “more” delayed the discovery of mathematical categories and, in turn, Conceptual Mathematics). Additional explanation of motion involves stating the even more obvious: rock remains rock during its motion. More broadly, every change of any object [of a category] preserves its essence (as alluded to earlier). For example, the transformation of young Posina into old Posina preserved Posina, and hence is a natural transformation (or ‘Becoming consistent with Being’, which is the Zeroth Law of Motion that Newton failed to abstract 😉 see also Lawvere and Rosebrugh, 2003, pp. 135, 173, 235-236, 241; Lawvere and Schanuel, 2009, pp. 152, 369-370, 378). Simply put, aging didn’t tear me apart (ibid, p. 146, 210).
Is there a major outstanding scientific problem that can be addressed using Conceptual Mathematics/Category Theory/Theory of Naturality or ಪರಿಕಲ್ಪನಾ ಅಧ್ಯಯನ (Parikalpane Adhyayana)? Yes, there is (see Fodor, 1998a; Núñez et al., 2019). In fact, category theory has already been applied to model various scientifically intractable notions such as the self that are encountered in consciousness studies (Ehresmann and Vanbremeersch, 2007; see also Comments on Reviews). Professor Andrée Ehresmann has been developing category theoretic constructs specifically designed to meet the demands of consciousness studies (Ehresmann, 1997, 2002). Category theory has also been of immense value in the foundational studies of automata/machines (Arbib and Manes, 1975, 1980), which can guide the development of a theoretical basis for the contemporary artificial/machine intelligence. In a true multidisciplinary spirit that our NEP aspires, Professor Michael Arbib made foundational contributions to many disciplines.
Reality consists of not only things and their motions and transformations, but also their reflections in our minds (as ideas or concepts) and consciousness (as percepts; Posina, 2019, 2020a; see also Lawvere, 1994a, 2004; Lawvere and Schanuel, 2009, pp. 84-85). Thus, we need a Category of Reflecting (with adjoint functors as objects; see ibid, pp. 369-377; Lawvere, 2016; Celebrating Bill Lawvere and Functorial Semantics; Posina, 2020b), in addition to the more familiar Categories of Being (or unity/cohesion objectified as reflexive graphs; Lawvere, 2007; see also Qualia Space) and Becoming (or change/variation objectified as dynamical systems; see Lawvere and Schanuel, 2009, pp. 135-141), to scientifically reconstruct reality by synthesizing ontology and epistemology into which reality is analyzed (cf. Posina, 2020c).
The kinship between science and cognition has been recognized time and again by scientists working in diverse disciplines such as mathematics, philosophy, and physics. For example, “science is nothing more than a refinement of everyday thinking” (Einstein, 1936/2003, p. 23; see also Fodor, 2006, p. 93; Lawvere, 2013a; Posina, 2020b; Posina, Ghista, and Roy, 2017; Schapira, 2016). When viewed from the perspective of this propinquity between scientific knowing and ordinary cognition, it is clear that we now have a good opportunity to build cognitive science (Fodor, 1998b) based on the “precise mathematical model for a very general scientific process of concept formation” (p. 7; see also Lawvere, 2002, pp. 267-268; Lawvere and Rosebrugh, 2003, pp. 235-236; Lawvere and Schanuel, 2009, pp. 127-129, 380).
More than anything else, we need intellectual investment! We need to invest in learning, teaching, and translating Conceptual Mathematics/Category Theory/Theory of Naturality into ಪರಿಕಲ್ಪನಾ ಅಧ್ಯಯನ (Parikalpane Adhyayana) so that we too can contribute our share to the world of science as envisioned in our education reforms (NEP, pp. 45-46). Translating Conceptual Mathematics (Lawvere and Schanuel, 2009) into ಪರಿಕಲ್ಪನಾ ಅಧ್ಯಯನ (Parikalpane Adhyayana) appears all the more significant in light of the ideal assessment of one’s understanding:
Write in your own words!
Most important of all, we wholeheartedly thank Professor Manjul Bhargava–on behalf of all Indian students, which include us–for taking his valuable time to draft our National Education Policy.
Originally published as:
Posina Venkata Rayudu Balla and Shankar (2021) Universal yearning for understanding, Science (eLetter, 01 August).
References
Arbib, M. A. and Manes, E. G. (1975) Adjoint machines, state-behavior machines, and duality, Journal of Pure and Applied Algebra 6: 313-344. https://www.sciencedirect.com/science/article/pii/0022404975900286
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Let us begin with something familiar: gift-bags. Gift-bags, as we all know, come in all shapes and sizes, so our definition of gift-bag, if it were to be good enough to describe each and everything that we think of as a gift-bag, should not involve any specification of a particular shape, or size, or color, to list a few properties. What, then, should the definition of gift-bag involve? Something—some property—that is true of all gift-bags, which is nothing more than the fact that they contain gifts. To the extent that there is nothing more to a gift-bag than the gifts that it contains, we might as well identify a gift-bag with its collection of gifts. How about gifts? Well, each gift is something that is distinct so that we can grab it from amongst other gifts in the collection of gifts, and yet is indistinguishable from other gifts in the sense that it is just a gift—nothing more, nothing less—just like every other gift in the set of gifts. This is all (to be qualified later) that we need to put in place the foundations of the monument called math.
A set is a collection of elements, and elements are distinct AND indistinguishable; a contradiction that confused many mathematicians when Cantor introduced his set of elements, with elements defined as lauter Einsen meaning ‘all ones’.
Here is a familiar contradiction (which might help with the unbearable burden of being a famous mathematicians finding it difficult to think, having long habituated to following rules 😉
Boundary(a) = a AND NOT(a)
For example, if we denote a region, say, India with i, then NOT(i) would be its neighbors Bangladesh, Bhutan, China, Nepal, Pakistan, Tibet, etc. Now, i AND NOT(i) is the boundary of India, or the border between India and its neighbors (no wonder we have so many border disputes; it is high time we make peace with contradictions 🙂
For reasons methodological (cf. finite), saving the “remote past” for later, we begin with Frege’s mathematical mistake:
CONCEPT = SET (of properties)
Our current scientific understanding of everyday human experience—a list of properties is not all that that we think of when thinking of one or another concept—can be geometrically objectified as a category, all of whose objects partake in the essence that is characteristic of that category; as such, transformations of objects (of the category) are natural (in the sense of ‘Becoming consistent with Being’; zeroth law of motion that Newton failed to abstract 😉
Admittedly, I find Professor F. William Lawvere’s Functorial Semantics fascinating (to the point of evangelism 😉
When Saunders Mac Lane penned his hard-hitting 1997 Synthese article, he was defending mathematics from an attack many of us hoped would just go away. But Saunders was aware of the seriousness of the threat, which indeed is still here with greater determination. Although the title of that article was “Despite physicists, proof is essential in mathematics”, he was not opposing physics, nor even that immediate handful who, assuming the mantle of “mathematical physicists”, gave themselves license to insult generations of scrupulously serious physicists and to demand that mathematics adopt a culture that considers conjecture as nearly-established truth. In essence it was an attack on science itself, as the highest form of knowing, that Saunders was opposing.
The increased determination of that attack is expressed in two ways. To equip and organize the attack, finance capital has set up several institutions, some of which rather openly proclaim their goal of submitting science to the service of medieval obscurantism. Others say that they support mathematical research, but encourage a barrage of “popular” writings to shock and awe the public into continuing in the belief that they will never understand mathematics and hence never be able to actively participate in science.
The contempt for Mac Lane’s fight, recently expressed in articles supposedly memorializing him, takes the form of the claim that category theory itself is a “cool” instrument for deepening obscurantism. Not only Harvard’s “When is one thing equal to another thing?” and the Cambridge “morality” muddle, but also a 2003 article aimed at teachers of undergraduates, quite explicitly support that claim. In the MAA Monthly, a Clay Fellow states as fact that category theory “is mathematics with the substance removed”. Mastering the technique of disinformation whereby the readers are first told that now finally they will be informed, the article suggests that some raising of the level of understanding of the relationship between space and intensively variable quantity is going to be achieved. Then the author short-circuits any such understanding via the simplifying assumption that omits the distinction between covariant and contravariant functors as “unwieldy”. As final display of the mastery of expositional technique, the categorical object which has, for nearly twenty pages, been heralded as simple, is revealed in the final pages in the most complicated and unexplained form possible. (Totally passed over is the issue that had led Grothendieck to the considerations allegedly being treated: not only the category of affine schemes, but also the category of all its presheaves, where the author implicitly wants us to work, fails to have the geometrically correct colimits needed to define projective space.)
Another level of attack was launched when Cornell University was given very large sums of money to develop methods of teaching geometry without mentioning any geometrical concepts. No proof of the desirability of such a draconian excising of content needed to be given, beyond some phrases from the Dalai Lama.
“Dumbing down” is an attack not only on school children and on undergraduates, but also one taking measured aim at colleagues in adjacent fields and at the general public. The general public is thirsty for genuinely informational articles to replace the science fiction gruel served constantly by journals like the Scientific American and the New York Times “Science” section. Those journals have never published anything resembling a mathematical proof and hence have rarely actually explained any scientific subject in a usable way. Nor have they even undertaken any program to raise the level of knowledge of calculus or linear algebra among their readers in a way which would make such explanations feasible. Instead, they provide games and amusements to divert the mathematically-interested public.
In January of 2005 the Notices of the AMS announced that they had for a full ten years been strictly following a certain editorial policy. There had been a widespread demand for expository articles. To that demand, the response was a new definition of “expository”: all precise definitions of mathematical concepts must be eliminated. Authors of expository articles were forced to compromise their presentation, or to withdraw their paper. Mathematicians, who were for several years becoming aware that these new expository articles are absolutely useless for developing a mathematical thought, were shocked to learn that a conscious policy had forced that situation.
A peculiar sort of anti-authoritarianism seems to be the only justification offered for degrading the role of definition, theorem, and proof; certainly, serious expositors have never considered that the use of those three pillars of geometrical enlightenment excludes explanations and examples. Others have urged, however, that those instruments be eliminated even from lectures at meetings and from professional papers.
That threat is part of the background for the concern expressed in the many messages to the categories list over the past weeks. Deeply concerned mathematicians ask me “How can we know?”. Indeed, how can we know whether it is worthwhile to attend a certain meeting or a certain talk, and how can a scientific committee know whether a proposed talk is scientifically viable? If the “you don’t want to know” culture of no proofs, no definitions, is accepted, we will truly have no way of knowing, and will be pressured to fall back on unsupported faith.
9 February 1937 – 23 January 2023
Do the exercises in the Conceptual Mathematics textbook; and mathematical understanding develops slowly.
Don’t expect establishments to enlighten 😉
9 February 1937 – 23 January 2023
Inter-Disciplinary Conference, Windsor, February 10, 1996
For a long time calculus teachers have been demanding better textbooks. In recent months, after much fanfare, a few books claiming to respond to this demand have been launched. But many teachers are saying already that this “new” is very similar to the old that worked so poorly, except now with much more and elaborate and subtle attempts to persuade the student (and the teacher) that some kind of “understanding” is going on, even while all the more resolutely concealing the essential principles.
Also in recent months, a journal for the guidance of college math teachers carried a lead editorial aiming to refute the “myth of scientific literacy”. Under the banner of this “scientific literacy” a major curriculum-restructuring over the past decades had been justified, which many scientists judged to be a forced…
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Conceptual Mathematics
Conceptual Mathematics 