Tags / 20 quotes from 7 sources
#mathematics
Beyond Zero and One (Andrew Smart)
The mind as a symbol-manipulating machine
Since Turing’s time, cognitive science has also assumed the mind to be a symbol-manipulating machine, an analogy to the abstract mathematical Turing machine. But as the example of computers that can play chess illustrates, how a computer does math and how a human does math may…
Concepts describe the world, experience is of it
One may be aware of one’s concepts and mental operations but that is the result of one’s capability of being conscious of the world (and thus also of certain internal tools to describe the world). In other words, we should not mix the concepts that we use to describe the world…
Infinitely many ways to represent numbers
Our concept of the number three can be represented as “3” in base ten, but as “0011” in binary. It is the same idea as using the word “three,” three fingers, or “III” to represent the concept of three. We likely use base ten to count because we have ten fingers and it naturally…
Leibniz: propagating faith through science
But Leibniz’s real reasons for wanting a binary numbering system were much deeper than the philistine goal of counting money. Leibniz, like almost all scientists of his day, was also deeply religious. He dreamed of inventing a universal language of logic that could once and for…
Leibniz withdraws his binary essay
Leibniz came up with binary numbers in about 1679, when he wrote a letter to a colleague about a “new science of numbers” on which he was working. Leibniz was soon invited by the Royal Academy of Sciences in Paris to contribute to their new annual publication, the Histoire de…
Penrose the mathematical Platonist
For all intents and purposes, Penrose is a Platonist when it comes to mathematical truth. This might seem at odds with his view that Gödel’s theorem shows that the mind in noncomputable. Penrose accepts the implications of Gödel’s theorem, which implies that human mathematical…
The robustness of mathematical truth
the precise mathematical entities from the approximations that we see around us in the world of physical things. In mathematics, we find a far greater robustness than can be located in any particular mind. Mathematics itself indeed seems to have a robustness that goes far…
Dehaene studies mathematics in the brain
Compared to Penrose, Dehaene’s approach to the issues of consciousness and math is much more empirical. Dehaene uses the tools of cognitive psychology and the latest brain imaging techniques to isolate precise brain regions, and to analyze exactly what these regions are doing…
Mathematical models are not reality
Edge magazine: Many mathematicians are Platonists. They think that the Universe is made of mathematical stuff, and that the job of mathematicians is merely to discover it. I strongly deny this point of view. This does not mean, however, that I am a ‘social constructivist’ … [I…
Is mathematics discovered or invented?
Penrose believes that math is precise while the physical world is approximate; Dehaene believes that math is an approximation of the physical world. Penrose believes that the universe is mathematically comprehensible and fundamentally mathematical, and further that mathematics…
Number sense is like color perception
Dehaene further likens our perception of math to how we perceive color. Color is not an objective property of the world that exists independently of human brains. Color does not exist in the world. Our brains create color when certain wavelengths of light hit our eyes, and they…
Ontologically subjective yet epistemically objective
Searle states, “Observer relativity implies ontological subjectivity but ontological subjectivity does not preclude epistemic objectivity.” To translate this into English: the existence of things that exist only because we believe them to exist does not mean that these things…
Gödel and the limits of AI
Roger Penrose argues that because humans can grasp Gödel’s theory, the mind is not a formal logical system. He argues that the source of our intuitions and consciousness must come from a new kind of quantum physics. Leonid Perlovsky says that within AI the consequences of…
Introducing Artificial Intelligence (Henry Brighton and Howard Selina)
Penrose's Gödelian argument against computable thought
To support his argument, Penrose appeals to a fundamental theorem in mathematical logic – Gödel’s Theorem – which states that certain mathematical truths cannot be proved by using a computational procedure. Because human mathematicians evidently can arrive at these truths…
Look Down Go Down
Research problems
In 1900 the German mathematician David Hilbert delivered a lecture in Paris in which he laid out the problems he thought mathematics might deal with in the coming century. Before getting to them he said what he thought were the general criteria which mark a good one. He…
Research Fronts 2014 : 100 Top Ranked Specialties In The Sciences And Social
Fractional co-citation clustering
Garfield and Small both continued their research and experiments in science mapping over the decade and thereafter. In two papers published in 1985, Small introduced an important modification to his method for defining research fronts: fractional co-citation clustering. By…
The Book Of Dead Philosophers (critchley, Simon)
Pythagorean numbers and the music of the spheres
Pythagorean doctrines were bound by an oath of secrecy, so we know very little prior to the version of them that appears in Plato. These include a belief in the immortality and transmigration of the soul and the view that the ultimate reality of the universe consists in number…
The Ego Trick (baggini, Julian)
The physical is what fits physicists' equations
the only way physicists can really understand the universe is in the language of mathematics. The fundamental basis of physics is not atoms, or even subatomic particles, but equations. That is why, for all its mysteriousness, dark matter does have all the reality that is needed…
Theories of Childhood, Second Edition (Carol Garhart Mooney)
Conservation of number and the spread-out coins
Because preoperational children tend to believe what they see, they do not yet have a firm grasp of qualities belonging to the objects in their world. For example, they confuse “heavy” with “large.” Due to inexperience, most young children would initially be surprised that a…
Concrete operations and reversibility
When children enter Piaget’s stage of concrete operations at about age seven, many changes in their thought patterns are visible. At this age (usually from about seven through eleven or twelve) children possess the characteristic of reversibility, which allows them to reverse…