The problem is that, without a fourth element, namely, an act of the intellect enclosing, as it were, or pervading, as it were, all three other elements in a unified idea, the three elements have no meaningful connection. And unless they have a meaningful connection as constituents of one complex, then they have no meaning on their own. United they mean, apart they don’t mean.
Even if I posted a flashing bright red sign on the wall that explained the intended correlation between the hash marks, the numbers, and my mental counting (e.g., moving my lips, moving my eyes in abstract neural space, wiggling my fingers as I count, etc.), unless a visitor to my room added a fourth element, namely, an intellectually integrated idea of the sign-paper-writer complex, the elements of my counting, writing, and posting a sign would have no meaningful connection. And without that connection, they cease to become what I intend them to be. Without a knowledge of how they relate, there is no way to say my counting isn’t really a form of dancing, or that the hash marks aren’t a portrait of my mother, and so on. No concept can really be a concept without an overriding intellectual conception of what those concepts are manifesting at a formal level. To believe otherwise is to go head to head with Hume and the lose metaphysics as the grid in which empiricalia become theoretically significant, and is to ignore the problem of ‘grueness’ and related problems.
Although the writer (or some visitor) intellectually conceives of the connection between abstract numbers, visible hash marks, and human mental operations BY MEANS OF a mental operation (and, for that matter, may emphasize the connection by writing down his mental conceptions, and, if he liked, may even more strongly assert the connection by encoding it with a natural-number sequence), but that intellectual conception is not itself a mental operation. If it were, it would just be one more dangling element in the complex, which still lacked any intellectually meaningful relation to (or intentionality with respect to) the other elements. The intellectual conception is the form of the complex as it is variously dematerialized BY MEANS OF the hash marks and tabulations. Forming an intellectual conception by means of mental operations no more means the former is the latter than drinking a glass of water means the water becomes glass when I drink it, or that a sinusoidal wave on the surface of a lake is itself a sheet of water.
As St. Thomas says in, SCG 1, 3:
"For since the leading principle of all knowledge of any given subject-matter is an understanding of the thing's innermost being, or substance––according to the doctrine of the Philosopher, that the essence is the principle of demonstration––it follows that the mode of our knowledge of the substance must be the mode of knowledge of whatever we know about the substance. Hence if the human understanding comprehends the substance of anything, as of a stone or triangle, none of the points of intelligibility about that thing will exceed the capacity of human reason. But this is not our case with regard to God. The human understanding cannot go so far of its natural power as to grasp His substance, since under the conditions of the present life the knowledge of our understanding commences with sense; and therefore objects beyond sense cannot be grasped by human understanding except so far as knowledge is gathered of them through the senses."[Cum enim principium totius scientiae quam de aliqua re ratio percipit, sit intellectus substantiae ipsius, eo quod, secundum doctrinam philosophi demonstrationis principium est quod quid est; oportet quod secundum modum quo substantia rei intelligitur, sit modus eorum quae de re illa cognoscuntur. Unde si intellectus humanus, alicuius rei substantiam comprehendit, puta lapidis vel trianguli, nullum intelligibilium illius rei facultatem humanae rationis excedet. Quod quidem nobis circa Deum non accidit. Nam ad substantiam ipsius capiendam intellectus humanus naturali virtute pertingere non potest: cum intellectus nostri, secundum modum praesentis vitae, cognitio a sensu incipiat; et ideo ea quae in sensu non cadunt, non possunt humano intellectu capi, nisi quatenus ex sensibilibus earum cognitio colligitur.]
In other words, paradoxically, the intellect is that which brings us "beyond" sensible, particular things to the immaterial, universal essences of them; yet, since our intellect depends, in this current mode of existence, on the senses, we cannot go "beyond" sensible things by sheer intellection alone. St. Thomas says explicitly (ST Ia, q, 84, a. 7, resp.) that
"In the present state of life in which the soul is united to a passible body, it is impossible for our intellect to understand anything actually, except by turning to the phantasms.
"First of all because the intellect, being a power that does not make use of a corporeal organ, would in no way be hindered in its act through the lesion of a corporeal organ, if for its act there were not required the act of some power that does make use of a corporeal organ [i.e., the neural, sensible basis of the intellect is a necessary but not SUFFICIENT basis for its operation]. …
"Now the reason of this is that the power of knowledge is proportioned to the thing known. Wherefore the proper object of the angelic intellect, which is entirely separate from a body, is an intelligible substance separate from a body. Whereas the proper object of the human intellect, which is united to a body, is a quiddity [i.e., a specific 'whatness'] or nature existing in corporeal matter; and through such natures of visible things it rises to a certain knowledge of things invisible. Now it belongs to such a nature to exist in an individual, and this cannot be apart from corporeal matter: for instance, it belongs to the nature of a stone to be in an individual [i.e., particular] stone, and to the nature of a horse to be in an individual horse, and so forth. Wherefore the nature of a stone or any material thing cannot be known completely and truly, except in as much as it is known as existing in the individual. Now we apprehend the individual through the senses and the imagination. And, therefore, for the intellect to understand actually its proper object, it must of necessity turn to the phantasms in order to perceive the universal nature existing in the individual."[Respondeo dicendum quod impossibile est intellectum nostrum, secundum praesentis vitae statum, quo passibili corpori coniungitur, aliquid intelligere in actu, nisi convertendo se ad phantasmata. Et hoc duobus indiciis apparet.
Primo quidem quia, cum intellectus sit vis quaedam non utens corporali organo, nullo modo impediretur in suo actu per laesionem alicuius corporalis organi, si non requireretur ad eius actum actus alicuius potentiae utentis organo corporali. …
Huius autem ratio est, quia potentia cognoscitiva proportionatur cognoscibili. Unde intellectus angelici, qui est totaliter a corpore separatus, obiectum proprium est substantia intelligibilis a corpore separata; et per huiusmodi intelligibilia materialia cognoscit. Intellectus autem humani, qui est coniunctus corpori, proprium obiectum est quidditas sive natura in materia corporali existens; et per huiusmodi naturas visibilium rerum etiam in invisibilium rerum aliqualem cognitionem ascendit. De ratione autem huius naturae est, quod in aliquo individuo existat, quod non est absque materia corporali, sicut de ratione naturae lapidis est quod sit in hoc lapide, et de ratione naturae equi quod sit in hoc equo, et sic de aliis. Unde natura lapidis, vel cuiuscumque materialis rei, cognosci non potest complete et vere, nisi secundum quod cognoscitur ut in particulari existens. Particulare autem apprehendimus per sensum et imaginationem. Et ideo necesse est ad hoc quod intellectus actu intelligat suum obiectum proprium, quod convertat se ad phantasmata, ut speculetur naturam universalem in particulari existentem.]
To relate this to analogy I proposed, I would replace "substance" with "intellectual conception of the mentally observed complex", and "the points of intelligibility" with "'mindable' elements of the complex." I would add that the disjunctive holds, to wit, if the human understanding does NOT comprehend the substance of anything, then any and all of the points of intelligibility about that thing will exceed the capacity of human reason.
Lest I forget this, for my own reference:"…it is necessary to say that the intellect is a power of the soul, and not the very essence of the soul. For then alone the essence of that which operates is the immediate principle of operation, when operation itself is its being: for as power is to operation as its act, so is the essence to being. But in God alone His action of understanding is His very Being. Wherefore in God alone is His intellect His essence: while in other intellectual creatures, the intellect is power."
–– ST Ia, q. 79, a. 1, resp.[…necesse est dicere, secundum praemissa, quod intellectus sit aliqua potentia animae, et non ipsa animae essentia. Tunc enim solum immediatum principium operationis est ipsa essentia rei operantis, quando ipsa operatio est eius esse, sicut enim potentia se habet ad operationem ut ad suum actum, ita se habet essentia ad esse. In solo Deo autem idem est intelligere quod suum esse. Unde in solo Deo intellectus est eius essentia, in aliis autem creaturis intellectualibus intellectus est quaedam potentia intelligentis.]