# Crisis? What Crisis? The Case For Neo-Intuitionism in Formal Science, Natural Science, and Philosophy, #2–What is Neo-Intuitionism?

# By Robert Hanna

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**TABLE OF CONTENTS**

**II. What is Neo-Intuitionism?**

**III. Putting Neo-Intuitionism To Work**

**IV. Conclusion**

**REFERENCES**

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You can also download and read or share a .pdf of the complete text of this essay–including the REFERENCES–**HERE**.

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**II. What is Neo-Intuitionism?**[i]

Neo-intuitionism says:

All semantic content, truth, and knowledge in the formal sciences, natural sciences, and philosophy,* are irreducibly partially determined by acts of rational human cognitive construction*, where “cognitive construction” is

either (i) the spontaneously creative *generation* *of legible texts* — whether by means of inner speech (thinking), outer speech (talking), or inscription (writing) — in some or another human natural or formal language L,

or (ii) the spontaneously creative *reading* *of legible texts* in some or another human natural or formal language L,

or (iii) the spontaneously creative *generation of essentially non-conceptual mental imagery* that’s indissolubly fused with (i) or (ii),

or (iv) the spontaneously creative *interpretation of essentially non-conceptual mental imagery* that’s indissolubly fused with (i) or (ii),

although at the same time, semantic content, truth, and knowledge in the formal sciences, natural sciences, and philosophy,* are not completely determined by acts of rational human cognitive construction alone*, due to the relatively independent contributions *also* made to them by

(v) normatively-governed human communities or *social institutions*, and

(vi) the manifestly real *external world*.

An essential point to notice here, is that the very act or process whereby a rational human animal *generates a legible text* presupposes that the text-generating person is *already able to read*, at the very least, *that legible text itself*: therefore, *reading legible texts logically and cognitively precedes generating legible texts*, even if children typically learn to talk before they learn to read.

In view of that essential point, then *what’s a legible text?*, and *what’s the act or process of reading*? In the interests of full philosophical disclosure, for the purposes of answering those questions, I’m presupposing

(i) the very ideas of

(ia) *a language, *including its characteristic syntactic and semanticproperties,and

(ib)* our knowledge of a language*, including our knowledge of its characteristic syntactic and semanticproperties (see, e.g., Chomsky, 1957, 1988),

(ii) a certain theory of *linguistic cognition *and* logical cognition* (see, e.g., Hanna, 2006b: esp. chs. 4 and 6),

(iii) a specifically *dual-content* cognitive semantics of *conceptual content* and *essentially non-conceptual content*, the latter of which also crucially functions as the source of what Otto Paans and I call *thought-shapers* (see, e.g., Hanna, 2015: esp. chs. 2 and 4; and Hanna and Paans, 2021), for the explanation of linguistic meaning, and above all, another necessary condition of reading:

(iv) the rational human capacity *to understand at least one language, at least minimally* (see, e.g., Wittgenstein, 1953; Chomsky, 1957, 1988).

Now, according to the *Oxford Encyclopedic English Dictionary*, “character” is defined as

*a printed or written letter, symbol, or distinctive mark. *(Hawkins and Allen, 1991: p. 247)

In view of that, and assuming that characters exist in inner speech (thinking), outer speech (talking), and inscription (writing) alike, then I’ll define a *text* as *any sequence of one or more characters*, where a one-character sequence is the lower-bound limiting case, and there’s no upper bound on the number of characters. In turn, what I’ll call a *text-in-L *is defined as *any sequence of one or more characters belonging to a particular human natural or formal language L*. It’s important to note that a language L can contain some characters (hence also some texts) that belong to one or more different languages L2, L3, L4, etc. So, for example, English contains some letters, words, and sentences belonging to other languages, including Greek, Latin, French, German, Italian, etc.

Then, I’ll provide necessary and sufficient conditions for legibility in two steps, as follows:

1. A text T-in-L is legible if and only if T-in-L satisfies *the perceptibility condition*, *the syntactic condition*, and *the semantic condition*, and

2. all and only such texts-in-L have legibility.

The perceptibility condition says that *the basic orientable (i.e., intrinsically directional) spatial shape and structure of T-in-L must be at least minimally perceptually detectable*, i.e., that T-in-L must be *at least* *partially perceptually detectable*, hence *it’s not completely perceptually undetectable*, and thereby T-in-L is *able-to-be-scanned* to at least that minimal extent. For example, if a text is completely blacked out, erased, otherwise completely smudged out or obscured, invisibly small, or so big that its shape cannot be perceived, then it’s perceptually undetectable and illegible. But on the other hand, as it were, even if a text T-in-L is right-to-left<–>left-to-right mirror-reversed and turned upside down, like this one in English —

it’s still *able-to-scanned* to the minimal extent that *it’s* *not completely undetectable*; and indeed, with a little effort, one can see that in fact it’s an upside-down enantiomorph of the extremely interesting English sentence I’ve elsewhere (Hanna, 2022e: section 5) dubbed *The Lector Sentence* —

**1. You, the reader of this very sentence,**

**can’t either coherently or self-consistently**

**deny that it’s self-evidently true that**

**you’re reading this very sentence.**

in explicit comparison-&-contrast with the classical *Liar Sentence*:

**This very sentence is false.**

The Lector Sentence and The Liar Sentence are

(i) each of them *reflexive*, i.e., self-referring,

(ii) each of them *self-manifesting*, and

(iii) mutually *antithetical*.

More specifically, The Lector Sentence is reflexive, non-contradictory, true, and furthermore self-manifestingly true, whereas The Liar Sentence is reflexive, contradictory, self-manifestingly false and paradoxical, and furthermore both true and false, i.e., a *truth-value glut*. In these ways, The Lector Sentence shows us *the foundations* of all science, truth, sound proof, and knowledge, whereas, as Tarski so brilliantly showed, The Liar Sentence shows us *the limits* of all science, truth, sound proof, and knowledge (Tarski, 1943, 1956).

In any case, the syntactic condition says that *T-in-L must be at least minimally well-formed*, i.e., that T-in-L must be *at least partially well-formed*, hence *it’s not completely ill-formed*, and thereby T-in-L is *able-to-be-parsed* to at least that minimal extent. For example, even if a text T-in-L is perceptually detectable, it can be completely jumbled, completely misspelled, or completely ungrammatical, or its characters can be completely randomly distributed, and in any of those ways it would be syntactically illegible. Indeed, ciphers or secret codes (as opposed to hidden messages in otherwise legible texts) are designed to *approach* syntactic illegibility, on the working assumption that the more illegible they are, the harder they are to break; so if there are some ciphers that *have never been broken* and all their creators are dead, or, more thought-experimentally, if there were a cipher created by intelligent non-human aliens that, even in principle, *could never be broken by rational human animals*, then they would be illegible in the syntactic sense. Therefore, a text-in-L’s satisfying the perceptibility condition, as such, *is not itself independently sufficient for readability* and thus *it’s not itself independently sufficient for being the target of any actual or possible act or process of reading*.

And the semantic condition says that *the conceptual content and/or essentially non-conceptual content of T-in-L must be at least minimally coherent*, i.e., that the conceptual content and/or essentially non-conceptual content of T-in-L must be* at least partially coherent*, hence *not completely incoherent*, and thereby the conceptual content and/or essentially non-conceptual content of T-in-L is *able-to-be comprehended* to at least that minimal extent. For example, even if a text is minimally perceptible and also minimally well-formed, nevertheless it can still violate minimal requirements of conceptual *sortal correctness* and/or essentially non-conceptual *sortal correctness*, or be strictly non-referential, and be semantic *gibberish*, hence be illegible in the semantic sense, like this non-poetical text-in-English, a paradigm case of sortal incorrectness, devised by Bertrand Russell (Russell, 1940: p. 166) —

*quadruplicity drinks procrastination*

or this famous poetical text-in-English, a paradigm case of strict non-referentiality, taken from Lewis Carroll’s *Jabberwocky* —

*’Twas brillig, and the slithy tovesDid gyre and gimble in the wabe;All mimsy were the borogoves,And the mome raths outgrabe.* (Carroll, 1988)

Therefore, that text from *Jabberwocky*’s satisfying the perceptibility condition *together with* the syntactic condition, yet also failing the semantic condition, shows that the first two conditions *are not themselves conjointly sufficient for readability* and thus that *they’re not themselves conjointly sufficient for being the target of any actual or possible act or process of reading*. Of course, millions of people, including you, the reader of this very sentence, have in some sense or another “read” that text from *Jabberwocky*; but my way of explaining away this apparent inconsistency is just to point out that *Jabberwocky* is indeed *legible* in both the perceptible and synactic senses (so in two senses, readable), but *illegible* in the semantic sense (so in one sense, unreadable), hence not legible in *all* relevant senses, hence *illegible* by my contextual definition, or conceptual analysis, of legibility. The same point holds, mutatis mutandis, for “quadruplicity drinks procrastination” and all other essentially similar texts-in-L: you can “read” it in two senses (the perceptible sense and the syntactic sense), but strictly speaking, it’s *illegible* according to the necessary and sufficient conditions of legibility, precisely because it fails the semantic condition.

Assuming all of that, I’m now in a position to provide precise necessary and sufficient conditions for the act or process of reading. In the following contextual definition, or conceptual analysis, by *person* I mean *rational human animal*: namely, a living human organism that’s capable of

(i) consciousness,

(ii) self-consciousness,

(iii) all forms of cognition including sense-perception, memory, and imagination, conceptualization, judgment, inference, theorizing, and a posteriori or a priori knowledge,

(iv) affect or emotion, and

(v) free will and practical agency (see, e.g., Hanna, 2015, 2018a, 2018b, 2018c).

Then, I’ll provide necessary and sufficient conditions for reading in two steps, as follows:

1. A person P reads a text T-in-L if and only if P consciously or self-consciously at least minimally scans, at least minimally parses, and also at least minimally comprehends T-in-L, and

2. all and only such acts or processes are reading.

It’s important to note that, consistently with this contextual definition, or conceptual analysis, of reading, a person P can read a text T-in-L either *aloud* or *silently to themselves*. It’s also important to note that neither scanning, nor parsing, nor comprehending, need be *self-consciously* or *reflectively* performed: this can be done in a more-or-less or even altogether *pre-reflectively* or *unself-consciously conscious* way; indeed, we typically “look right through” what we’re reading in order to go directly to the *meaning* (whether sense, reference, or speech-act uptake) of what we’re reading, and altogether overlook the scanning, parsing, and comprehending dimensions of the act or process of reading itself. In order to bring those dimensions back into view, all you have to do is to repeat any text-in-L — for example, a sentence or word — out loud a few times (say, ten times) until it sounds strangely *bereft* of meaning; that strange *absence-of-meaning* has then become vividly manifest to you precisely because the perceptibility and syntax of that particular text-in-L have been temporarily self-consciously detached from what you’ve been previously been pre-reflectively and unself-consciously yet still consciously comprehending. And it’s also important to note that the point I made above about “readers” of *Jabberwocky* and “quadruplicity drinks procrastination” goes, mutatis mutandis, for my contextual definition, or conceptual analysis, of reading: of course, millions of people, including you, the reader of this very essay, are in some sense or another “readers” of that text from *Jabberwocky*; and no doubt a few thousand people have read “quadruplicity drinks procrastination”; but my way of explaining away this apparent inconsistency too, is just to point out that *Jabberwocky* and “quadruplicity drinks procrastination” can indeed be *read* in both the perceptible and synactic senses (so in two senses, that’s reading), but *cannot be read* in the semantic sense (so in one sense, that’s *not* reading), hence it’s not reading in *all* the relevant senses, hence it’s *not reading* by my contextual definition, or conceptual analysis, of reading.

As we’ve just seen, the intentional targets of the act or process of reading are at-least minimally scannable, at-least minimally parse-able, and at-least minimally comprehensible *structural objects belonging to some or another human natural or formal language L*, that are ineluctably embedded in an egocentrically-centered, intrinsically directional or orientable, manifestly real, three-dimensional space, thereby necessarily requiring the actual existence and essential embodiment of the reader. As linguistic structural objects, the intentional targets of reading are manifestly real *linguistic physical tokens* of manifestly real *linguistic physical types*, which in turn are *inherently repeatable objects* that are *non-platonically and kantianly abstract* according to this definition:

*X is non-platonically and kantianly abstract if and only if X **is not uniquely located and realized in manifestly real spacetime**, and X is concrete otherwise.* (Hanna, 2015: pp. 269–270)

Now, the *rational human cognition* of concrete tokens of the linguistic structural objects of reading, whether in perception, memory, or imagination, is what Kant calls *sensibility* (*Sinnlichkeit*), which in turn requires a capacity for first-order conscious or self-conscious, essentially non-conceptual, and non-empirical *unified formal spatial or temporal representation*, or what Kant calls *pure intuition *or* reine Anschauung *(Kant, 1998: p. 173, A20/B34–35). Therefore, the act or process of reading is an essentially *intuitionistic* activity *that doesn’t require any sort of platonic objects*. Neo-intuitionism thereby wholly avoids the classical metaphysical/ontological and epistemic problems of platonism, especially including *The Benacerraf Dilemma*, which says:

(i) on the one hand, our standard Tarskian semantics of mathematical truth requires platonically abstract objects that exist outside of spacetime and are causally inert, but

(ii) on the other hand, our best theory of human knowledge requires directly sensibly accessible causal objects of perception, so

(iii) mathematical truth is humanly unknowable. (Benacerraf, 1973)

In short, the act or process of reading, by virtue of its intuitionistic nature, is decisively (to coin a nifty neologism) *trans-Benacerraf-Dilemma-istic*, precisely because it’s metaphysically *structuralist*, ontologically *non-platonistic*, although fully accommodating *non-platonically and kantianly abstract objects*, and epistemically *skepticism-resistant*, from the get-go (Hanna, 2015: chs. 6–8).

If we add the analyses of legibility and reading, and the decisively trans-Benacerraf-Dilemma-istic character of the act or process of reading, to the general definition of neo-intuitionism, then we have in front of us a synoptic overview of what neo-intuitionism *is *and* entails, *and also an initial survey of what neo-intuitionism *substantively presupposes*. But, in addition to that initial survey, what *else* does neo-intuitionism substantively presuppose? Neo-intuitionism also presupposes six other substantive theses or theories, as follows.

**First**, neo-intuitionism presupposes *sensible set-theory*, which says:

*ZFC must be restricted to all and only the actual or possible objects of rational human sensibility.* (Hanna, 2022a: appendix 4)

**Second**, neo-intuitionism presupposes *rational anthropology for uncomputable functions*, which says:

*for every uncomputable or undecidable function whatsoever in the formal or natural sciences, in order to explain the effective operations of that function beyond uncomputability or undecidability, **we postulate an innately-specified rational human capacity, or a unified set of such capacities, for effectively performing that function.** *(Hanna, 2022f)

**Third**, neo-intuitionism presupposes *Kantian structuralism*, which says:

*(1) The natural numbers are essentially positions or roles in the mathematical natural number structure provided by Peano Arithmetic in its full generality and denumerable infinitude, beyond the denumerable finitary sub-structure provided by Primitive Recursive Arithmetic, also including ontologically robust, non-denumerable, and impredicatively constructed conservative extensions of Peano Arithmetic such as Cantorian Arithmetic.*

*(2) The Löwenheim-Skolem theorem, together with the Upward Löwenheim-Skolem theorem proved by Tarski, collectively show that Cantorian Arithmetic is a conservative extension of Peano Arithmetic, especially including Primitive Recursive Arithmetic, by showing*

*(i) that a first-order mathematical theory has non-denumerably infinite models if and only if it has denumerably infinite models, and*

*(ii) that a first-order mathematical theory has denumerably infinite models only if it has denumerably finite models.*

*(3) The mathematical natural number structure provided by Peano Arithmetic (and Primitive Recursive Arithmetic and Cantorian Arithmetic) is abstract only in the non-platonic, Kantian sense that it is weakly or counterfactually transcendentally ideal. This is the same as to say that this structure is identical to the structure of the Kantian “formal intuition” of time — as an iterative sequence of homogeneous units that is inherently open to the primitive recursive functions — as we directly and veridically cognize it in Kantian pure or a priori intuition, as represented by formal autonomous essentially non-conceptual content. This content, in turn, must be taken together with all the formal concepts and other logical constructions, including specific logical inference patterns such as mathematical induction, needed for an adequate rational human comprehension of Peano Arithmetic (and Primitive Recursive Arithmetic and Cantorian Arithmetic), that we cognize through conceptual understanding or thinking.*

*(4) In our actual world, the unique, intended model of the non-platonic, Kantian abstract natural number structure provided by Peano Arithmetic (and Primitive Recursive Arithmetic and Cantorian Arithmetic) is just an immanent structure that is fully embedded in the set of manifestly real, directly and veridically perceivable spatiotemporal material objects in nature. This immanent structure determines how these material natural objects are the role players of the Peano Arithmetic-(and-Primitive Recursive Arithmetic-and-Cantorian Arithmetic)-specified natural number roles in the non-platonic, Kantian abstract formal structure of time as we directly and veridically cognize it in Kantian pure or a priori intuition, as represented by formal autonomous essentially non-conceptual content. This content, in turn, must be taken together with all the formal concepts and other logical constructions, including specific logical inference patterns such as mathematical induction, needed for an adequate rational human comprehension of Peano Arithmetic (and Primitive Recursive Arithmetic and Cantorian Arithmetic), that we cognize through conceptual understanding or thinking.* (Hanna, 2015: pp. 381–382)

**Fourth**, neo-intuitionism presupposes *Kantian intuitionism*, which says:

*[Phenomenologically self-evident, luck-resistant, skepticism-resistant] a priori knowledge in mathematics, by means of basic authoritative mathematical rational intuition, is the joint product of two distinct yet closely coordinated cognitive capacities*.

*(1) On the one hand, mathematical intuition flows from a rational human animal’s capacity for generating, scanning, reproducing, and manipulating schematic mental imagery that is also veridical, in the dual sense that (i) it correctly maps onto its intentional targets, and (ii) those targets really exist. In Kantian terms, this imagery is constituted by sensible forms given in pure or a priori intuition, constructed by the productive imagination. This capacity is innately specified in the rational human animal’s mind as a cognitive competence, and it is also inherently present, as a necessary ingredient, in all rational human sense perception. Mathematical intuition thus entails the rational human animal’s self-conscious and self-reflective cognition of phenomenologically self-evident formal structures of object-directed and self-directed sense perception.*

*(2) And on the other hand, mathematical intuition also flows from a rational human animal’s capacity for constructing logics and natural languages. This capacity is innately specified in her mind as a cognitive competence, and also it is inherently present, as a necessary ingredient, in all rational human empirical conceptualizing and perceptual judgment. Mathematical intuition thereby also entails the rational minded animal’s self-conscious and self-reflective cognition of phenomenologically self-evident formal conceptual contents and specific patterns of logical inference in classical or non-classical logics. *(Hanna, 2015: p. 394)

**Fifth**, neo-intuitionism presupposes *weak or counterfactual transcendental idealism*, which says:

*(i) Things-in-themselves/noumena are logically possible, but at the same time it is knowably unknowable and unprovable whether noumena/things-in-themselves exist or not, hence for the purposes of an adequate anthropocentric or “human-faced” metaphysics, epistemology, and ethics, they can be ignored (= **radical agnosticism and methodological eliminativism about noumena/things-in-themselves**).*

*(ii) Necessarily, all the proper objects of rational human cognition have the same forms or structures as — i.e., they are isomorphic to — the forms or structures that are non-empirically generated by our innately-specified spontaneous cognitive capacities, but at the same time those manifestly real worldly forms or structures are not literally type-identical to those a priori cognitive forms or structures (= **the isomorphism-without-type-identity thesis**.*

*(iii) It’s a necessary condition of the existence of the manifestly real world that if some rational human animals were to exist in that world, then they would veridically cognize that world, via either autonomous essentially nonconceptual content or conceptual content, at least to some extent, at least to some extent (= **the counterfactual cognizability thesis**).*

*(iv) The manifestly real world has at some earlier times existed without rational human minded animals, or any other minded beings, to know it, and could exist even if no rational human minded animals, or any other minded beings, ever existed to cognize it veridically, even though some rational human animals now actually exist in that manifestly real world (for example, I [R.H.]), who do in fact cognize it veridically, at least to some extent (= **the existential thesis**).*

*Here’s a slightly more precise formulation of Weak or Counterfactual Transcendental Idealism’s crucial thesis (iii), the counterfactual cognizability thesis:*

*Syn Ap □ (∀x) (∃y) [MRWx → {(RHAy & MRWy) □→ VCyx}]*

*Definitions:*

*Syn Ap □ = synthetically a priori necessarily*

*P □→ Q = If P were the case, then Q would be the case*

*MRWx = x belongs to the manifestly real world*

*MRWy = y belongs to the manifestly real world*

*RHAy = y is a rational human animal*

*VCyx = y veridically cognizes x, at least to some extent = either y veridically cognizes x via autonomous essentially non-conceptual content or y veridically cognizes x via conceptual content, at least to some extent.*

*Natural Language Translation:*

*Synthetically a priori necessarily, anything that belongs to the manifestly real world is such that if some rational human animals were to exist in that world, then they would veridically cognize that thing, at least to some extent, via either autonomous essentially non-conceptual content or conceptual content.*

*Two Crucial Implications:*

*(1) The counterfactual knowability thesis holds even if **no **rational human minded animals, or any other minded beings, actually exist, or ever actually existed.*

*(2) If anything is such that rational human minded animals are unable to cognize it veridically, via autonomous essentially non-conceptual content or conceptual content, at least to some extent — for example, things-in-themselves or noumena — then that thing does not belong to the manifestly real world.*

*Crucial implication (1) conveys the weak mind-independence and ontic integrity of the manifest[ly real] world. The manifest[ly real] world is what it is, even no minds actually exist or ever actually existed. And crucial implication (2) conveys the weak mind-dependence and inherent knowability of the manifestly real world. The manifest[ly real] world is what it is, only in relation to actual or possible minds like ours. The single upshot of the two crucial implications is that the manifest[ly real] world is as real as anything can ever possibly be, on the reasonable assumption that some luck-resistant, skepticism-resistant rational human knowledge of that world is actual or really possible. *(Hanna, 2015: pp. 337–338)

**Sixth**, finally, and above all, neo-intuitionism presupposes *the neo-organicist worldview*, centered on the root metaphor of *the living organism* (for example, a plant, an animal, and above all, rational, self-conscious minded animals like us), which says:

*(i) that everything in the world is essentially or fundamentally uncomputable, negentropic, processual, purposive, self-organizing, time-irreversible or time–asymmetric, and non-equilibrium thermodynamic,*

*(ii) that there is a basic metaphysical and ontological continuity, running from the Big Bang singularity to uncomputable, negentropic, time-asymmetric, non-equilibrium thermodynamic energy flows, to living organisms, to conscious minded animals, to rational, self-conscious minded animals with free will and practical agency, and finally to social institutions of all kinds (Torday, Miller Jr, and Hanna, 2020; Hanna, 2022a), and*

*(iii) that all mechanical systems whatsoever, whether formal or natural, are **nothing but systematic abstractions from** — that is, degenerate cases of, fragments of, or limiting cases of — fundamentally organic systems, and therefore all mechanical systems whatsoever are **logically or nomologically supervenient on** organic systems *(Hanna, 2022a: esp. ch. 4 and Appendices 1–4).

Before indicating how neo-intuitionism can be applied to the mega-crisis in the formal and natural sciences and post-classical Analytic philosophy, it’s instructively useful to compare and contrast neo-intuitionism with *classical* intuitionism:

*Intuitionism is a philosophy of mathematics that was introduced by the Dutch mathematician L.E.J. Brouwer (1881–1966). Intuitionism is based on the idea that mathematics is a creation of the mind. The truth of a mathematical statement can only be conceived via a mental construction that proves it to be true, and the communication between mathematicians only serves as a means to create the same mental process in different minds.*

*This view on mathematics has far reaching implications for the daily practice of mathematics, one of its consequences being that the [logical] principle of the excluded middle, (A∨¬A), is no longer valid. Indeed, there are propositions, like the Riemann hypothesis, for which there exists currently neither a proof of the statement nor of its negation. Since knowing the negation of a statement in intuitionism means that one can prove that the statement is not true, this implies that both A and ¬A do not hold intuitionistically, at least not at this moment. The dependence of intuitionism on time is essential: statements can become provable in the course of time and therefore might become intuitionistically valid while not having been so before.*

*Besides the rejection of the principle of the excluded middle, intuitionism strongly deviates from classical mathematics in the conception of the continuum, which in the former setting has the property that all total functions on it are continuous. Thus, unlike several other theories of constructive mathematics, intuitionism is not a [conservative] restriction of classical reasoning; it contradicts classical mathematics in a fundamental way. *(Iemhoff, 2020, bracketted material and underlining added; see also Brouwer, 1981, 1999)

I*ntuitionistic logic is an offshoot of L.E.J. Brouwer’s intuitionistic mathematics. A widespread misconception has it that intuitionistic logic is the logic underlying Brouwer’s intuitionism; instead, the intuitionism underlies the logic, which is construed as an application of intuitionistic mathematics to language. Intuitionistic mathematics consists in the act of effecting mental constructions of a certain kind. These are themselves not linguistic in nature, but when acts of construction and their results are described in a language, the descriptions may come to exhibit linguistic patterns. Intuitionistic logic is the mathematical study of these patterns, and in particular of those that characterize valid inferences. An inference rule is valid if, whenever the statements in the premises describe truths of intuitionistic mathematics, a construction can be found that makes true the statement that is obtained by applying the rule. What the principles of logic need to preserve is therefore not, as in classical logic, mind-independent truth, but mental constructibility. Various principles of classical logic, most notably the Principle of the Excluded Middle, then become insufficiently grounded, and certain classical theorems even contradictory. The theorems in intuitionistic logic that formally contradict classical theorems depend on elements of intuitionistic mathematics that are incompatible with classical mathematics; this illustrates how in intuitionism logic is based on mathematics and not the other way around. *(van Atten, 2022, underlining added; see also Brouwer, 1981, 1999)

Thus classical intuitionism, as per Brouwer, says that semantic content, truth, and knowledge in *all* of mathematics and *parts* of logic, are *wholly* based on inherently *individualistic* mental acts of *non-linguistic* rational human cognitive construction, a fact that entails *the rejection of the principle of excluded middle*. More precisely, classical intuitionism is

(i) *restricted to* mathematics and parts of logic,

(ii) *psychologistic* or even *solipistic*,

(iii) *non*-linguistic, and

(iv) committed to *non-classical or deviant logic*, and more specifically to a rejection of the principle of excluded middle,

whereas neo-intuitionism is *none* of these, and indeed neo-intuitionism *specifically rejects all of (i) to (iv)*. More precisely, neo-intuitionism

(i*) has* unrestricted application *to all the formal and natural sciences, and all of philosophy,

(ii*) is *anti*-psychologistic and *anti*-solipsistic (Hanna, 2006: ch. 1),

(iii*) is inherently *language-centered* (Hanna, 2006: ch. 4), and

(iv*) is fully committed to *the minimal principle of non-contradiction* — i.e., necessarily and a priori, not every sentence is both true and false (Putnam, 1983; Hanna, 2006: ch. 2, 2015: ch. 5) — and also to its De Morgan equivalent, *the minimal principle of excluded middle* — i.e., necessarily and a priori, some sentences are either true or false with no third value and no value-gap, i.e., necessarily and a priori, not every sentence is neither true nor false with a third value or a value-gap.

**NOTE**

[i] Parts of this section are adapted from (Hanna, 2022c: sections 6 and 00).

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