Category Archives: knowledge modeling

What are numbers, really?

The nature of math came up in our embodied cognition discussion. Here is a presentation by Dehaene on the topic followed by comments from The Reality Club: George Lakoff, Marc Hauser, Jaron Lanier, Rafael Núñez, Margaret Wertheim, Howard Gardner, Joseph Traub, Steven Pinker, Charles Simonyi. A few brief, edited Lakoff excerpts follow from that discussion. Note that this is a discussion from 1997, so a lot of confirming science has happened since then.

” [Dehaene] has made it clear that our capacity for number has evolved and that the very notion of number is shaped by specific neural systems in our brains. […] We understand the world through our cognitive models and those models are not mirrors of the world, but arise from the detailed peculiarities of our brains.”

“Mathematics is not ‘abstract’, but rather metaphorical, based on projections from sensory-motor areas that make use of ‘inferences’ performed in those areas. The metaphors are not arbitrary, but based on common experiences: putting things into piles, taking steps, turning around, coming close to objects so they appear larger, and so on.”

“Dehaene is right that this requires a nonplatonic philosophy of mathematics that is also not socially constructivist. Indeed, what is required is a special case of experientialist philosophy (or ’embodied realism’). […] Such a philosophy of mathematics is not relativist or socially constructivist, since it is embodied, that is, based on the shared characteristics of human brains and bodies as well as the shared aspects of our physical and interpersonal environments. […] On the other hand, such a philosophy of mathematics is not platonic or objectivist. Consider two simple examples. First, can sets contain themselves or not? This cannot be answered by looking at the mathematical universe. You can have it either way, choosing either the container metaphor or the graph metaphor, depending on your interests.”

“Dehaene is by no means alone is his implicit rejection of the Computer Program Theory. Distinguished figures in neuroscience have rejected it (e.g., Antonio Damasio, Gerald Edelman, Patricia Churchland). Even among computer scientists, connectionism presents a contrasting view. In our lab at the International Computer Science Institute at Berkeley, Jerome Feldman, I, and our co-workers working on a neural theory of language, have discovered results in the course of our work suggesting that the program-mind is not even a remotely good approximation to a real mind.”

Pinker of course also comments and takes issue with Lakoff’s depiction of him. Dehaene also responds to the comments at the end.

Should quantum anomalies make us rethink reality?

Indeed we should according to this recent Scientific American article. One thing I learned from Dennett in his new book is that according to him the scientific image uncovers an objective, underlying reality, while the manifest image is corrupted by our personal ontology. Not so according to quantum mechanics, which operates on the premise that our scientific results themselves are tied to our perceptions and constructed categories, not “a purely objective world out there.” There is a paradigm shift in science itself in accepting this understanding, given quantum anomalies described in the link. It’s time to update your scientific (and manifest) image Professor Dennett.

Query on interest in concept mapping

I like using concept maps to organize my thoughts and to plan articles and projects. I would like to share member-editable concept maps through this website.

I’ve used the free Cmap Tools program for many years. Unfortunately, Cmap Tools is developed on the Java platform and I have been unable to get it to work on my Mac for about two years. (Apparently, it needs an older version of the Java Runtime Environment and several other tools I use need a later version, which creates a conflict I’ve not been able to resolve.)

I am able to use Cmap Tools on my Windows 10 virtual machine, so it’s still a good candidate application.

Fortunately, there’s also a free cloud version of Cmap we can use to share concept maps. The downloadable (client) application has many more features for embedding functionality into concept maps, but the cloud environment is excellent for viewing and basic editing. Here’s a sample cmap I built in about five minutes this morning (displayed using the embed code from the Cmap Cloud Viewer):

Because this example is presented in a viewer window, it is not directly editable. If I provide a link to a shared cmap online, you’ll be able to edit it as well. If you install the downloadable version of Cmap Tools, you’ll be able to add richer content and more advanced features, including collapsable detailed nodes, embedded images and other resources, links to other nodes and maps, links to online content, etc.

If you might be interested in viewing and/or collaborating on concept maps, please reply and identify the operating system of the computer you would use. (If you don’t want to post your OS reply, please tell me by email.) Once you create a free account online, we’ll be able to collaboratively work on cmaps, which is a great way to negotiate meaning with a bit more structure and persistence than real-time conversations.

Last, if you want to know how cmaps and other visual representation benefit learning, read the articles on the IHMC site. I also highly recommend the little book, Learning How to Learn, by Novak and Gowin. Thanks!