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Neural Mechanics�Week 2: Concepts as Vectors�and Steering

Tuesday, January 20, 2026

David Bau

Northeastern University

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Today’s plan: lots to do

Team S, E research pitches.

Talk about the reading:

What is a concept?
What is the big deal with vectors?
Why is binding and composition hard?
What is steering doing?

Aruna will present: Notebook on steering

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Team S and E

Research pitches

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Vectors can’t bind variables or�compose systematically�(Fodor & Pylyshyn 1988)

Symbols can't explain graded�similarity, typicality, or learning�(Rosch, Rumelhart)

Amodal symbols have no meaning�— the grounding problem� (Harnad)

Modal simulations can't explain�abstract concepts (justice, seven, if)

Multimodal vectors can incorporate�perceptual grounding

Vectors trained on text along are ungrounded�— meaning isn’t relational position

2017+:�Transformers,�LLMs

1988:�Fodor & Pylyshyn�critique

1980s:�PDP connectionist�revival

1950-1960s:�Symbolic AI�Chomsky’s revolution

1990s:�Embodied�cognition emerges

Isaac: Concept? Christopher: Falsifiable? Jasmine: vs Symbols? Haoyu: Superposition feature? Kai: models as vectors?

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Vectors can’t bind variables or�compose systematically�(Fodor & Pylyshyn 1988)

Symbols can't explain graded�similarity, typicality, or learning�(Rosch, Rumelhart)

Amodal symbols have no meaning�— the grounding problem� (Harnad)

Modal simulations can't explain�abstract concepts (justice, seven, if)

Multimodal vectors can incorporate�perceptual grounding

Vectors trained on text along are ungrounded�— meaning isn’t relational position

2017+:�Transformers,�LLMs

1988:�Fodor & Pylyshyn�critique

1980s:�PDP connectionist�revival

1950-1960s:�Symbolic AI�Chomsky’s revolution

1990s:�Embodied�cognition emerges

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Juggling Three Perspectives on Concepts

What is a concept?

Example:

“The food that Alice loves”��Question: what is this concept?

Neural state

“vector”

Computational�“role”

Referenced “thought”

A function mapping all�food to the probability�that Alice loves it

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How many animal concepts in 5 neurons?

Localist encoding? (“grandmother neuron”)

5 animals, in any combination

Combinatorial encoding? (“bitstring of neurons”)

2^5=32 animals, but only one

Geometric encoding? (“vector of neurons”)

If orthogonal/independent: just 5 animals.

Relax independence: dozens of “almost independent” animals

Rice: More features than dimensions?

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Warmup: Elhage’s Toy Data Experiment

Problem: remembering the random size x_i (0..1) of 20 animals.
But we only have 5 neurons to use.
Some animals more important to remember than others, (0.7)ⁱ.
Often animals are absent, P[x_i = 0] = S

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Dot products can encode more structure

Orthogonality:

densely�independent concepts have zero dot product.

“Superposition”:

sparse and independent concepts can have nonzero dot product

Courtney, Avery: Superposition stable? Yuqi: overlap? Luze: inherently unsafe? Claire: sparsity?

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Three Views of Dot Products

* Matching * Angles * Decomposition

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The weirdness of high dimensions

	Three Dimensions	1000 Dimensions
The dot product of two random unit vectors	0.46	0.025
Volume of 99% radius ball	97%	0.004%
How much of the unit cube is filled by the unit sphere	52%	10^-2700
Normal distribution, avg \|x\|	1.7	31.6
Normal distribution std \|x\|	40%	3%

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Why Binding is Hard

Jesseba: Adhoc composition? Arya: why is binding hard?

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Binding solution explodes dimension

VSAs solve this by reducing the dimension, with some noise.
LLMs can bind, but do LLMs actually do this? We don’t know. Ask Nikhil!

See: Smolensky 1990�Tensor Product Variable Binding

Horse

Fedora

Cat

Cap

Yunus: Doesn’t LLM composition vindicate? Ayush: shared concepts multitask?

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ITI Steering Towards “Truthful” Answers

Claire, Ananya: belief vs output? Yiqian: hallucination vs lying? Grace: TruthfulQA? Armita: prompting orthogonal?

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Lab activity

Explore vectors

try steering

(Aruna)