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Missing Layers II
#MissingLayer #GSMSymbolic #MathCognition #LLM #Neuroscience
This dissociation is not a peculiarity of advanced mathematics. It is present from birth. The intraparietal system — the approximate number system, the system that estimates quantities, compares magnitudes, and represents spatial relations — is active in infants before language acquisition and is present in non-human primates. It is an evolutionarily ancient circuit, operating below and independently of the linguistic channel. Einstein stated: 'Words and language, whether written or spoken, do not seem to play any part in my thought processes.' The neuroscience confirms this is not idiosyncrasy. It is the normal architecture of mathematical cognition in minds capable of it: the intraparietal system does the mathematical work, and language serves as the external scaffolding that translates mathematical structure into communicable symbolic form.
The cultural evidence reinforces this. The Munduruku of the Amazon have number words only to five, after which they use approximate terms. The Pirahã, also Amazonian, have no exact number words at all — only approximations for 'few' and 'many'. Both peoples can perform approximate quantity comparisons correctly, using the intraparietal system that does not require language. What they cannot do, without the linguistic scaffolding of exact number words, is perform the precise arithmetic that formal mathematics requires. The language is not the mathematics. The language is the interface that connects the intraparietal system to the symbolic precision that makes mathematics formal and communicable. Remove the language and the approximate intuition remains. Remove the intraparietal system and you have language about mathematics without the mathematical substance it describes.
A large language model has neither. It has no physical grounding, no approximate number system, no spatial reasoning substrate, no evolutionarily ancient circuit for quantity and magnitude. What it has is the linguistic interface: the pattern of how mathematical structure is described in human language, the surface form of how problems are posed and solutions are expressed, the statistical regularities of mathematical discourse in its training corpus. The GSM-Symbolic study shows exactly how far that interface extends and exactly where it ends: it extends through the familiar linguistic patterns of the training distribution and ends when the numerical surface changes while the mathematical structure is preserved.
The implication for the series' central argument is formally precise. The GSM-Symbolic study shows a a fundamental limitation: an LLM cannot reliably solve grade-school arithmetic when the numbers are changed, because it does not have the intraparietal layer that constitutes mathematical reasoning.
The human brain uses distinct neural systems for language and mathematics. The intraparietal system — evolutionarily ancient, present in infants and primates, dissociated from language areas — is what mathematical reasoning runs on. LLMs have neither this system nor the physical grounding layer. The GSM-Symbolic study shows the consequence: change the numbers in a word problem while keeping the mathematical structure identical, and performance drops across all models. They learned the language of mathematics. They did not learn mathematics. The interface without the substrate is not the thing itself.
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“Chimpanzees display remarkable mathematical abilities that parallel human cognition in fundamental ways. They understand simple quantities, can order numbers, and perform basic numerical operations such as addition. Particularly striking is their photographic short-term memory for briefly displayed number sequences — in controlled tests they consistently outperform adult humans on this task, likely because humans rely more heavily on verbal processing, which turns out to be a disadvantage here.“
Missing Layers I
#MissingLayer #GSMSymbolic #MathCognition #LLM #Neuroscience
The sixth post, The Missing Layer described the layer that LLMs do not have: the physical grounding layer through which the body embeds concepts in physical reality. The eagle reading thermals, the cat landing on its feet, the fish sensing current with its lateral line — these are computations that do not pass through the linguistic channel. They arise from direct sensorimotor contact with the physical world, encoded in neural circuits that predate language by hundreds of millions of years.
A study published by Apple researchers in 2024 — GSM-Symbolic — provides the most precise empirical demonstration yet of what this missing layer costs in the domain of mathematical reasoning. The researchers created variants of standard mathematical word problems by changing only the numerical values, keeping the linguistic structure and logical form identical. The finding was unambiguous: the performance of every tested language model declined significantly when only the numbers were changed, even though the mathematical structure of the problem was unchanged. A model that appeared to solve the original problem failed on the variant. The performance drop was not small. It was consistent across all models tested, from the largest available systems to smaller ones.
This result has a precise interpretation. A system that genuinely reasons mathematically would perform equally well on structurally identical problems regardless of the specific numerical values, because the mathematical structure — not the surface pattern of the numbers — is what determines the solution. A system that pattern-matches on the linguistic form of mathematical problems would perform well on problems similar to its training data and poorly on variants that shift the numerical surface while preserving the mathematical structure. The GSM-Symbolic results show the second pattern. The models are not doing mathematics. They are recognising the linguistic pattern of mathematical problem descriptions and generating the linguistic pattern of solutions. When the numbers change, the surface pattern changes, and the pattern-matching fails.
fMRI studies of professional mathematicians have established that mathematical reasoning in the human brain is not processed through the language areas. When mathematicians evaluate the truth of advanced mathematical statements — in algebra, analysis, topology, geometry — the activation is in bilateral intraparietal sulci and inferior temporal regions. These are the regions associated with number perception, quantity comparison, and spatial reasoning. Crucially, these activations spare the areas associated with language and general semantic knowledge. The mathematical and linguistic networks are dissociated. They use different cortex. They relate to different neuronal networks. They are, in the formal sense of the series, different layers.
Picture by ChatGPT, Claude helped to express my thoughts in words.
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