Tag: Computation

An Embodied Mathematics

Posted on by Herbert Harris

by Herbert Harris

The Stepped Reckoner, a calculating machine invented by philosopher and mathematician Gottfried Wilhelm Leibniz in 1674

Is mathematics created or discovered? For over two thousand years, that question has puzzled philosophers and mathematicians alike. In Plato’s Meno, Socrates encourages an uneducated boy to “discover” a geometrical truth simply by answering a series of guided questions. To Plato, this demonstrated that mathematical knowledge is innate, that the soul recalls truths it has always known. The intuitionists of the early twentieth century, however, rejected this idea of eternal forms. For thinkers like Poincaré and Brouwer, mathematics was not revelation but construction: an activity of the human mind unfolding in time.

The debate continues today in an unexpected new arena. As artificial intelligences start to generate proofs, conjectures, and even entire branches of formal reasoning, we are prompted to ask again: what does it mean to do mathematics? Current systems excel at symbol manipulation and pattern matching, but are they truly thinking in any meaningful way, or just rearranging signs? The deeper question is how humans do mathematics. What happens in the brain when a mathematician recognizes a pattern, intuitively sees a relation, or invents a new kind of number?

In what follows, I’ll trace that question from ancient philosophy to modern neuroscience and then to the newest foundations of mathematics. We’ll see that mathematical invention may be the natural expression of the brain’s recursive, embodied intelligence, and that this perspective could transform how we think about both mathematics and AI. Read more »

Monday, May 31, 2021

To See A World In A Grain Of Silicon: Why Minds Aren’t Programs

Posted on by Jochen Szangolies

by Jochen Szangolies

Figure 1: The IBM 5150 Personal Computer, introduced August 1981. Image credit: wikipedia.

The year 2021 marks the 40th anniversary of the introduction of IBM’s first Personal Computer (PC), the IBM 5150. Since then, computers have risen from a novelty to a ubiquitous fixture of modern life, with a transformative impact on nearly all aspects of work and leisure alike.

It is perhaps this ubiquity that prevents us from stopping to ponder the essentially mysterious powers of the computer, the same way a fish might not ponder the nature of the water it is immersed in.

By ‘mysterious powers’, I don’t mean the impressive capabilities modern computers offer, in terms of, say, data storage and manipulation—while it is no doubt remarkable that even consumer grade devices today are able to beat the best human players at chess, and the engineering behind such feats is miraculous, there is nothing mysterious about this ability.

No, what is mysterious is instead the feat of computation itself: a computer is, after all, a physical object; while a computation, say something straightforward like calculating the sum of two numbers, operates on abstract objects. Therefore, the question arises: how does the computer qua physical system connect to abstract objects, like numbers? Does it reach, somehow, into the Platonic realm itself? To the extend that computers can use the result of computations to drive machinery, they seem to present a bridge by which the abstract can have concrete physical effects. Read more »