# Reasoning Babies, Abstract Principles and Probabilities

It happens many times in class that I say, “in mathematics when you see something you don’t know, you try to figure it out using something you do know.  And, recently, in the context of thinking about the generalizations that blossomed in late 19th and early 20th century mathematics, I’ve also wondered how it is that we stay ‘on track’ so to speak.  How is it, for example that Riemann’s foundation for geometry holds onto the working properties of the original ideas?

I think that both of these things are partially addressed by recent research in cognitive science.  In this post, I’d like to bring your attention to the work of a team led by Josh Tenenbaum, at MIT.  It was recently reported that this team made the observation that babies reason – that they will expect an outcome of a situation based on a few physical principles.

These experiments make use of a tool that has been developed to measure a baby’s surprise, because this provides a way to identify a baby’s expectations.  Their surprise (or lack of it) is measured by the amount of time they look at something.  In other words, the baby will look at something longer when an unexpected thing has happened.  These times have been carefully and repeatedly recorded.  As Tenenbaum has said, researchers have quantified surprise.  The same kind of tool has been used to measure a baby’s number sense. But this particular study identifies reason based on principles, before there is language.  According to Tenenbaum, the study

suggests infants reason by mentally simulating possible scenarios and figuring out which outcome is most likely based on a few physical principles.

The report made me want to look more at Tenebaum’s work. I found a link to his recent article in Science:  How Minds Grow.  The link is on his web page under the heading Representative reading and talks.  The article begins with a question that I think can be applied directly to mathematics: How do our minds get so much from so little? But the article is a complex analysis of how we build our conceptual structures on probabilities. Early abstractions develop when we order the distinguishable features of a perceived object, and these develop with experience, opening vast territories of knowledge, with what Tenenbaum calls a hierarchical Bayesian framework. The body builds its world based on probabilities related to experience or evidence.  And conceptual systems grow in tree-like structures. The full content of the article is beyond the scope of this post.  But the work contributes to the current view that perception and understanding happen together, that their interaction is seamless.  Abstraction is a fundamental aspect not only of vision but of learning and all aspects of the body’s interaction with its environment. Perhaps the body ‘knows’ how to use abstraction the way it knows how to use light, for example.

In the conclusion of the article, Tenenbaum says the following:

How can structured symbolic knowledge be acquired through statistical learning? The answers emerging suggest new ways to think about the development of a cognitive system. Powerful abstractions can be learned surprisingly quickly, together with or prior to learning the more concrete knowledge they constrain. Structured symbolic representations need not be rigid, static, hard-wired, or brittle. Embedded in a probabilistic framework, they can grow dynamically and robustly in response to the sparse, noisy data of experience.

These observations give me a way to think about how mathematics stays on track and why intuition can play so crucial a role.  It may be that mathematics searches the paths taken by concepts (first rooted in the body’s managing physical experience with abstract principles) or just searches the possibilities for concepts, within the constraints (or principles) the body knows.  The ‘intellect’ and the senses are here united in a provocative way.

### 4 comments to Reasoning Babies, Abstract Principles and Probabilities

• My hunch (and the idea I’m most interested in) is that everything we have come to think of as ‘intelligent’ is some ongoing interweaving of fundamental nervous system talents (including mathematics), and that they happen because the body just continues to do what it does. The ‘why’ might just broaden our idea of what one means by ‘living.’

Thanks for visiting. I hope you come back.

• I think a really interesting question is *why* we are possibly born with an innate sense of reason. Why, evolutionarily, was that advantageous? I think there are obvious answers–such as how interpreting the world in a semi-objective, logical fashion leads to survival–but those obvious answers lead to questions about how animals interpret the world, and how much reasoning they are born with, and how their reasoning systems are different from ours.

• Hello Aviv,

I think you’re correct when you say that only the baby’s response is being measured, but there is a history now in cognitive science of using this kind of measurement to get a handle on very early expectations. Like with anything, finding precisely discernible patterns in the observations is what would make them useful, or even reliable. (I first saw it when Stanislas Dehaene used it to identify a number sense in babies) I’m not a cognitive scientist, so I don’t know well the source of their confidence in the measurement. As for the ‘reasoning’ idea, I think their observations are being used to suggest that there is a fundamental aspect to reasoning, some cognitive structure, maybe even within sensory mechanisms, that grows into what we later call ‘reason.’ This may be my own take on it, but I think it would at least be consistent with how researchers see it.

Thanks for the comment. I’m so glad you visited!

Joselle

• Thanks Josselle,
I’m very happy to discover your blog and to find one (?) more person interested in math and embodiment.
Reading this post it seems to me that there are several gaps in Tenenbaum’s reasoning.
First of all, what he measures is behaviour (of the babies) and certainly not their emotional/mental state. Surprise implies a conscious refutal of expectations and I think it’s important to note that at least as you report it there is no evidence for that.
Secondly, the fact of the babies having expectations does not at all imply that these were born out of reasoning. Expectations can rise out of conditioning (for example). I’m sure my objection is one that Tenenbaum could anticipate so I wonder how he deals with it.

Aviv