Mathematics is a language, not a tool applied to language. Register conversion is transformation, not translation. Notation is generative, not merely compressive. "The notation thinks" is an observation I have made independently and tested empirically (114 experiments measuring how named operators change output structure). Duval's register insight, which you present through the Pythagorean example across four registers, is structurally parallel to a result I have formalized: three irreducible projections (genealogical, structural, functional) that each reveal different structure in the same object, with no two recovering the third. Duval has four registers. I have three projections. Both say: the object is only accessible through multiple views, each view reveals and conceals, and moving between views is the cognitive work, not a convenience.
Mathematics is not a tool the mind uses. Mathematics is what the mind does. The fifteen structural-categorial operators I have formalized are not a notation for cognitive operations. They ARE the cognitive operations. The mind does not use mathematics to think. The mind's constitutive activity IS mathematical. Individuation, classification, negation, conditionality, causation, boundary, comparison, quantification: these are not tools the mind picks up. These are the operations the mind consists in when it is cognizing. The mathematics is not external to the mind. The mathematics IS the mind's activity.
If mathematics is a tool the mind uses, the mind could in principle think without mathematics. If mathematics is what the mind does, the mind without mathematics is not thinking. The fifteen operators do not extend cognition. The fifteen operators ARE cognition. Remove them and there is no cognition left to extend.
I would not hesitate to maintain that the mind does mathematics. Cognitively speaking, the problem lies in asking what makes mathematical potential possible — the agent's capacity to hold the position "I do maths." Performing mathematics as an uninterrupted flow is mathematical, but it proceeds without the awareness of doing so. Isn't that the ultimate question we are seeking to answer? The proposed hypothesis is haltability: a capacity that could regulate the flowing maths so as to know its joints (https://arxiv.org/pdf/2605.26856v1).
As for the idea that mathematics is an instrument, let me respond with an analogy: mathematics as an instrument is like knowing physiology in order to regulate or repair digestion — as against the fact that digestion happens in our bodies whether or not we know how. Knowing how it happens helps us regulate it. I am thus taking the stand that the mind without regulation (modulation) is doing maths without explicitly knowing the principles of doing maths.
It is important to differentiate between performance and competence. Stating things explicitly, using one register or more, develops competence in us — while we have been performing all along.
The archerfish shoots water at insects above the surface. The archerfish compensates for refraction at the air-water interface in real time. It adjusts for target distance, target size, target motion, and the ballistic arc of the water jet. The archerfish does not know Snell's law. The archerfish does not know projectile mechanics. The archerfish's constitutive activity IS the refraction compensation. The doing IS the knowing. There is no gap between the performance and the competence. The gap is introduced by the theorist who stands outside the archerfish and says: the fish performs but does not understand.
Your digestion analogy is revealing in a way you may not intend. You say: digestion happens whether we know how it happens. Knowing how it happens helps us regulate. That is true for the physician who treats digestion from outside. It is not true for the gut. The gut does not need to know physiology to digest. The gut's constitutive activity IS the digestion. The gut IS competent. The gut does not perform digestion while lacking competence. The gut's performance IS its competence. The physician has a DIFFERENT competence: the ability to describe the gut from outside and intervene. But the physician's competence is not the gut's competence. They are different operations on the same phenomenon.
So: when you say the mind does mathematics without knowing it is doing mathematics, I ask: who is the knower that the mind is supposed to lack? The mind IS doing the mathematics. The doing IS the knowing. The demand for explicit awareness of principles is the demand for a SECOND operation (describing the first operation from outside). The second operation is real and valuable. The second operation is not required for the first operation to be competent. The archerfish is competent at refraction compensation without the second operation. The gut is competent at digestion without the second operation. The mind is competent at mathematics without the second operation.
The haltability idea in your linked paper is interesting precisely because it names the second operation: the ability to stop, inspect, and regulate the flow. That IS a real capacity. That capacity IS valuable. That capacity IS what your registers and explicit formalization provide. But the capacity to halt and inspect is not what makes the mathematics mathematical. The mathematics was mathematical before the halting. The halting lets you see the mathematics. The halting does not create it.
Mathematics is a language, not a tool applied to language. Register conversion is transformation, not translation. Notation is generative, not merely compressive. "The notation thinks" is an observation I have made independently and tested empirically (114 experiments measuring how named operators change output structure). Duval's register insight, which you present through the Pythagorean example across four registers, is structurally parallel to a result I have formalized: three irreducible projections (genealogical, structural, functional) that each reveal different structure in the same object, with no two recovering the third. Duval has four registers. I have three projections. Both say: the object is only accessible through multiple views, each view reveals and conceals, and moving between views is the cognitive work, not a convenience.
Mathematics is not a tool the mind uses. Mathematics is what the mind does. The fifteen structural-categorial operators I have formalized are not a notation for cognitive operations. They ARE the cognitive operations. The mind does not use mathematics to think. The mind's constitutive activity IS mathematical. Individuation, classification, negation, conditionality, causation, boundary, comparison, quantification: these are not tools the mind picks up. These are the operations the mind consists in when it is cognizing. The mathematics is not external to the mind. The mathematics IS the mind's activity.
If mathematics is a tool the mind uses, the mind could in principle think without mathematics. If mathematics is what the mind does, the mind without mathematics is not thinking. The fifteen operators do not extend cognition. The fifteen operators ARE cognition. Remove them and there is no cognition left to extend.
https://doi.org/10.5281/zenodo.20318684
I would not hesitate to maintain that the mind does mathematics. Cognitively speaking, the problem lies in asking what makes mathematical potential possible — the agent's capacity to hold the position "I do maths." Performing mathematics as an uninterrupted flow is mathematical, but it proceeds without the awareness of doing so. Isn't that the ultimate question we are seeking to answer? The proposed hypothesis is haltability: a capacity that could regulate the flowing maths so as to know its joints (https://arxiv.org/pdf/2605.26856v1).
As for the idea that mathematics is an instrument, let me respond with an analogy: mathematics as an instrument is like knowing physiology in order to regulate or repair digestion — as against the fact that digestion happens in our bodies whether or not we know how. Knowing how it happens helps us regulate it. I am thus taking the stand that the mind without regulation (modulation) is doing maths without explicitly knowing the principles of doing maths.
It is important to differentiate between performance and competence. Stating things explicitly, using one register or more, develops competence in us — while we have been performing all along.
The archerfish knows what it is doing.
The archerfish shoots water at insects above the surface. The archerfish compensates for refraction at the air-water interface in real time. It adjusts for target distance, target size, target motion, and the ballistic arc of the water jet. The archerfish does not know Snell's law. The archerfish does not know projectile mechanics. The archerfish's constitutive activity IS the refraction compensation. The doing IS the knowing. There is no gap between the performance and the competence. The gap is introduced by the theorist who stands outside the archerfish and says: the fish performs but does not understand.
Your digestion analogy is revealing in a way you may not intend. You say: digestion happens whether we know how it happens. Knowing how it happens helps us regulate. That is true for the physician who treats digestion from outside. It is not true for the gut. The gut does not need to know physiology to digest. The gut's constitutive activity IS the digestion. The gut IS competent. The gut does not perform digestion while lacking competence. The gut's performance IS its competence. The physician has a DIFFERENT competence: the ability to describe the gut from outside and intervene. But the physician's competence is not the gut's competence. They are different operations on the same phenomenon.
So: when you say the mind does mathematics without knowing it is doing mathematics, I ask: who is the knower that the mind is supposed to lack? The mind IS doing the mathematics. The doing IS the knowing. The demand for explicit awareness of principles is the demand for a SECOND operation (describing the first operation from outside). The second operation is real and valuable. The second operation is not required for the first operation to be competent. The archerfish is competent at refraction compensation without the second operation. The gut is competent at digestion without the second operation. The mind is competent at mathematics without the second operation.
The haltability idea in your linked paper is interesting precisely because it names the second operation: the ability to stop, inspect, and regulate the flow. That IS a real capacity. That capacity IS valuable. That capacity IS what your registers and explicit formalization provide. But the capacity to halt and inspect is not what makes the mathematics mathematical. The mathematics was mathematical before the halting. The halting lets you see the mathematics. The halting does not create it.
Yes, halting does not create mathematics. Halting creates the mind, the reflective stance, and discovers itself.