#mathematical constructivism

[pv1isalsoimportant]

mathematical revelation so great i almost became religious

[omegaphilosophia]

The Philosophy of Arithmetic

The philosophy of arithmetic examines the foundational, conceptual, and metaphysical aspects of arithmetic, which is the branch of mathematics concerned with numbers and the basic operations on them, such as addition, subtraction, multiplication, and division. Philosophers of arithmetic explore questions related to the nature of numbers, the existence of mathematical objects, the truth of arithmetic propositions, and how arithmetic relates to human cognition and the physical world.

Key Concepts:

The Nature of Numbers:

Platonism: Platonists argue that numbers exist as abstract, timeless entities in a separate realm of reality. According to this view, when we perform arithmetic, we are discovering truths about this independent mathematical world.

Nominalism: Nominalists deny the existence of abstract entities like numbers, suggesting that arithmetic is a human invention, with numbers serving as names or labels for collections of objects.

Constructivism: Constructivists hold that numbers and arithmetic truths are constructed by the mind or through social and linguistic practices. They emphasize the role of mental or practical activities in the creation of arithmetic systems.

Arithmetic and Logic:

Logicism: Logicism is the view that arithmetic is reducible to pure logic. This was famously defended by philosophers like Gottlob Frege and Bertrand Russell, who attempted to show that all arithmetic truths could be derived from logical principles.

Formalism: In formalism, arithmetic is seen as a formal system, a game with symbols governed by rules. Formalists argue that the truth of arithmetic propositions is based on internal consistency rather than any external reference to numbers or reality.

Intuitionism: Intuitionists, such as L.E.J. Brouwer, argue that arithmetic is based on human intuition and the mental construction of numbers. They reject the notion that arithmetic truths exist independently of the human mind.

Arithmetic Truths:

A Priori Knowledge: Many philosophers, including Immanuel Kant, have argued that arithmetic truths are known a priori, meaning they are knowable through reason alone and do not depend on experience.

Empiricism: Some philosophers, such as John Stuart Mill, have argued that arithmetic is based on empirical observation and abstraction from the physical world. According to this view, arithmetic truths are generalized from our experience with counting physical objects.

Frege's Criticism of Empiricism: Frege rejected the empiricist view, arguing that arithmetic truths are universal and necessary, which cannot be derived from contingent sensory experiences.

The Foundations of Arithmetic:

Frege's Foundations: In his work "The Foundations of Arithmetic," Frege sought to provide a rigorous logical foundation for arithmetic, arguing that numbers are objective and that arithmetic truths are analytic, meaning they are true by definition and based on logical principles.

Russell's Paradox: Bertrand Russell's discovery of a paradox in Frege's system led to questions about the logical consistency of arithmetic and spurred the development of set theory as a new foundation for mathematics.

Arithmetic and Set Theory:

Set-Theoretic Foundations: Modern arithmetic is often grounded in set theory, where numbers are defined as sets. For example, the number 1 can be defined as the set containing the empty set, and the number 2 as the set containing the set of the empty set. This approach raises philosophical questions about whether numbers are truly reducible to sets and what this means for the nature of arithmetic.

Infinity in Arithmetic:

The Infinite: Arithmetic raises questions about the nature of infinity, particularly in the context of number theory. Is infinity a real concept, or is it merely a useful abstraction? The introduction of infinite numbers and the concept of limits in calculus have expanded these questions to new mathematical areas.

Peano Arithmetic: Peano's axioms formalize the arithmetic of natural numbers, raising questions about the nature of induction and the extent to which the system can account for all arithmetic truths, particularly regarding the treatment of infinite sets or sequences.

The Ontology of Arithmetic:

Realism vs. Anti-Realism: Realists believe that numbers and arithmetic truths exist independently of human thought, while anti-realists, such as fictionalists, argue that numbers are useful fictions that help us describe patterns but do not exist independently.

Mathematical Structuralism: Structuralists argue that numbers do not exist as independent objects but only as positions within a structure. For example, the number 2 has no meaning outside of its relation to other numbers (like 1 and 3) within the system of natural numbers.

Cognitive Foundations of Arithmetic:

Psychological Approaches: Some philosophers and cognitive scientists explore how humans develop arithmetic abilities, considering whether arithmetic is innate or learned and how it relates to our cognitive faculties for counting and abstraction.

Embodied Arithmetic: Some theories propose that arithmetic concepts are grounded in physical and bodily experiences, such as counting on fingers or moving objects, challenging the purely abstract view of arithmetic.

Arithmetic in Other Cultures:

Cultural Variability: Different cultures have developed distinct systems of arithmetic, which raises philosophical questions about the universality of arithmetic truths. Is arithmetic a universal language, or are there culturally specific ways of understanding and manipulating numbers?

Historical and Philosophical Insights:

Aristotle and Number as Quantity: Aristotle considered numbers as abstract quantities and explored their relationship to other categories of being. His ideas laid the groundwork for later philosophical reflections on the nature of number and arithmetic.

Leibniz and Binary Arithmetic: Leibniz's work on binary arithmetic (the foundation of modern computing) reflected his belief that arithmetic is deeply tied to logic and that numerical operations can represent fundamental truths about reality.

Kant's Synthetic A Priori: Immanuel Kant argued that arithmetic propositions, such as "7 + 5 = 12," are synthetic a priori, meaning that they are both informative about the world and knowable through reason alone. This idea contrasts with the empiricist view that arithmetic is derived from experience.

Frege and the Logicization of Arithmetic: Frege’s attempt to reduce arithmetic to logic in his Grundgesetze der Arithmetik (Basic Laws of Arithmetic) was a foundational project for 20th-century philosophy of mathematics. Although his project was undermined by Russell’s paradox, it set the stage for later developments in the philosophy of mathematics, including set theory and formal systems.

The philosophy of arithmetic engages with fundamental questions about the nature of numbers, the existence of arithmetic truths, and the relationship between arithmetic and logic. It explores different perspectives on how we understand and apply arithmetic, whether it is an invention of the human mind, a discovery of abstract realities, or a formal system of rules. Through the works of philosophers like Frege, Kant, and Leibniz, arithmetic has become a rich field of philosophical inquiry, raising profound questions about the foundations of mathematics, knowledge, and cognition.

[d6b-onion]

But p≡¬(¬p)

Not necessarily!

Constructive logic is a framework where much greater emphasis is put on actually constructing the objects you use in proofs in the like.

In that sense, the law of excluded middle p v not(p) is interpreted as "i have either a proof of p or a proof of not(p)", which may not always be true. You could have a proof of neither!

And in constructive logic, 'not not p' is a weaker statement than p. p always implies 'not not p", but the opposite is not always true.

Hence why fanfiction discourse rejects the law of excluded middle.

[synthetic-ultramarine]

hmmm top 5 historical art movements you wish more people knew about?

Soviet Constructivism. an abstract, industrial soviet art and design movement. I think it is less well-known than socialist realism, but you get a clearer picture of the soviet art world if you know about both.

Egyptian Surrealism. My favorite art history professor covered the Egyptian surrealist movement in one of her classes. Look at this manifesto.

Old Kingdom portrait sculpture. The popular idea about ancient egyptian art is that depictions of the human figure were always simple, stylized, and highly idealized. In fact, there were artistic movements, and there were changes in how the figure was depicted over time. The Boston MFA has a collection of portrait busts from the old kingdom, from a moment when realism was in fashion; likenesses of people who lived 4000 years ago, their faces so lifelike that they almost seem to breathe.

Computer art. A decades-old field full of rich and varied works. Long before the AI marketing boom there was a history of weird and wonderful things being made by the small crowd of people who are interested in both postmodern art and graduate-level mathematics.

20th century sci-fi and fantasy Illustration. Somebody needs to do a serious art-historical examination of this. It might have to be me.

[wielsonf]

why qualitative research matters to teaching (mathematics)

the pioneer cohort of ed.d. and ph.d. educ at ateneo gbseald with dr. ma. assunta "achoot" cuyegkeng / screengrabs from markkie aribon and lavi subang of ed.d.

-----

i told the internet about why i am where i am now in a previous post. now, i am ready to share what invaluable knowledge i've got from one of my courses so far.

during my undergrad and graduate studies, just pen and paper is mostly sufficient to create new ideas, problems, and solutions, and do research in math, but that's not the only concern i have with life and work.

for the past half a decade or so, peers and i saw a decline in the perceived quality of students in calculus at the university of the philippines los banos. what was supposed to be tackled in senior high school, like algebra, trigonometry, and precalculus especially for those from stem track, wasn't adequately done so. when introduced to fundamental calculus concepts, they do understand the notion of limit, how lines and derivatives intertwine and entangle, and why area is an integral, but when it comes to crafting solutions and answering problems, this inadequacy becomes apparent.

in turn, it becomes necessary for us to ask why this is the case? how do teachers influence their students' calculus learning and what are they doing to adapt? has existing policies done us (dis)service in the philippine (math) education? and, why is everybody and nobody at fault here?

just thinking about possible solutions is not enough. we need to get our hands dirty, wreck some established norms, and possibly hurt some feelings in the process. qualitative research, i learned, has some of the answers.

as i mentioned in a previous post,

for this course, i posed this question: how do teachers facilitate collegiate calculus learning through creative interventions?

in an attempt to answer this question, i had to look back at what is happening and what research tells us about calculus learning.

for one, most mathematics learning theories are based on existing ones from classical learning theories, like constructivism, positivism, and behaviorism. since the start of the 21st century, one of the main motivations of mathematics education research is rationalizing a theory for mathematics in consideration of its unique nature as a subject matter.

next comes becoming aware of challenges and factors in teaching and learning calculus, how do teachers intervene, and how important their role is.

as of yet, i think the question can't be answered by a simple survey, answered using a likert scale. we have to go on the ground and diligently ask calculus teachers and instructors in college the existing literature still resonate with the challenges they face and what they do about them in order to help their students. the quality of administration and prestige of the institutions they serve affect their students, but in reality, educators are at the frontline facing the students and implementing interventions as they go along day by day. their experience is a vital part of understanding the problem.

we should break down such a complex question into easily digestible and directly answerable ones that help us understand sac (structures-agency-culture): ask where the teachers come from, what kind of pool do they dive in to teach calculus, what restrictions they are put in, what the students are like, what they do to help the children [sic], and in what way do their interventions affect the students.

with the pisa results just released, now more than ever, we need to act as fast as we can to implement changes from the ground up.

why are we, everyone at school, so miserable in one way or another? this, i can definitely say, is my magnum opus.

i will die on this hill.

[ghelgheli]

also while mathematical constructivism has its place (most of all in contexts where explicit construction is pragmatically useful) i think it fails on aesthetic terms. you don't like proofs by contradiction because they only claim to establish existence without delivering what is supposed to exist? tell me you don't believe in things you can't see. tell me you've never known something was there without touching it. the future will come even if we don't know what form it may take.

[ferricoasia]

Quote 76 Tell me and I forget, teach me and I may remember, involve me...

The Science Behind "Involve Me and I Learn": Why Active Participation Beats Passive Learning

"Tell me and I forget, teach me and I may remember, involve me and I learn." — Benjamin Franklin

Introduction

Benjamin Franklin's famous quote is more than a catchy phrase—it's a blueprint for effective learning and a powerful indictment of passive education. In just a few words, it highlights the limitations of traditional teaching and emphasizes the value of active engagement. Whether you're a teacher, parent, coach, or lifelong learner, this principle holds the key to meaningful education in any setting.

Stage One: "Tell Me and I Forget" Pitfall of Passive Learning

We've all experienced the fleeting impact of being told something. A lecture, a set of instructions, a YouTube tutorial — easy to absorb, but just as easy to forget. This is because passive learning barely scratches the surface of cognitive engagement.

In education, this is where the "chalk and talk" method often fails. It delivers content but rarely leaves a lasting imprint unless it's followed up with reinforcement or practice. Listening is not the same as learning.

Did You Know?

Studies show we retain only 5-10% of information from lectures compared to 75-90% from active practice (National Training Laboratories).

Stage Two: "Teach Me and I May Remember" – The Role of Structured Instruction

Teaching takes things a step further. A good teacher breaks down complex ideas, uses examples, and offers guidance. This level of explanation helps learners form mental associations that aid retention.

However, even structured teaching has its limits. Retention still relies heavily on the learner's ability to make the knowledge their own—by seeing relevance, applying it, or linking it to prior understanding.

It's a better approach than simply telling, but still incomplete without the final, crucial component.

Stage Three: "Involve Me and I Learn" – Learning Through Experience

True learning happens when individuals are actively involved. This could mean hands-on activities, problem-solving, collaboration, experiments, simulations, or real-world application.

Supported by Learning Theories:

Experiential Learning (Kolb): Learning is a process whereby knowledge is created through experience.

Constructivism (Piaget, Vygotsky): Learners construct knowledge through active engagement.

Project-Based Learning: Involvement through creation and inquiry leads to deeper understanding.

When people participate—rather than just observe—they internalize concepts, develop critical thinking, and gain confidence.

Real-Life Examples

A student learns mathematics better by building a budget for a real project than just solving textbook problems.

A trainee nurse becomes competent not from reading about injections but from administering them under supervision.

A child remembers road safety more effectively when crossing streets with a parent than from a cartoon video alone.

Involvement breeds competence, and competence builds confidence.

Why This Matters in a Digital World

In an age of information overload, passive learning is cheap and everywhere. You can find endless videos, articles, and eBooks on any topic. But real skill-building comes from doing.

Online platforms that integrate interactive modules, quizzes, discussion forums, and real-world tasks are far more effective than static content dumps. Similarly, workplaces that foster learning-by-doing outperform those that rely solely on onboarding videos or manuals.

Pro Tip:

When learning something new, follow the 70/30 rule: Spend 30% of your time consuming information and 70% applying it through practice.

Conclusion: Make Learning Stick

Franklin's quote is a wake-up call: stop lecturing. Start involving.

Whether you're educating children, training staff, or improving yourself, remember:

Telling is not enough.

Teaching is better.

Involving is where the magic happens.

If your goal is transformation—not just information—then involvement must be the foundation.

Ready to Apply Active Learning?

To watch on YouTube - https://youtu.be/Mmub5HgS7IM

Disclosure: This article may contain affiliate links to recommended resources. We only recommend products we believe in.

Tags: #ActiveLearning #ExperientialEducation #LearningByDoing #EducationReform #TeachingStrategies #LifelongLearning #SkillDevelopment

[teen--spirit]

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[geopolicraticus]

The power set of facts yields innumerable histories.

Constraining History by Constraining Facts.—The instability of facts—that the world can be decomposed into facts in various different ways—is another sense in which history is infinite (in addition to the sense in which we can construct an infinitude of formal facts about the past). As the natural numbers can be decomposed into more subsets than there are natural numbers, so too the events of history can be decomposed into innumerable facts, and these facts can be assembled into more histories than there are events in history. That no type theory was ever imposed upon facts, attempting to arrange them hierarchically according to their complexity or extent, is a fundamental ambiguity of science, though an understandable ambiguity in at least a couple of ways. It is understandable insofar as rectifying this ambiguity would likely undercut the results of science in a way much like the use of a constructive logic undercuts much classical mathematics—and for similar reasons. Non-constructive thought, or what Torkel Franzén called classical eclecticism, gives a free hand in methodology, whereas all forms of constructivism—intuitionism, predicativism, finitism—involve some principled constraints on methodology, and a constraint on method will ultimately mean a constraint on results. A type theory for facts would constrain the results of science. And the absence of a type theory of facts is further understandable insofar as we don’t possess the formal framework necessary to enforce a hierarchy upon facts. All type theories have become unmanageably complex. The type theory of Principia Mathematica required work-arounds like the axiom of reducibility, which seem flagrantly unnatural. (Mathematicians don’t use type theory, they use ZF or some similar axiomatization.) Some other formalization of facts, or their elimination, seems to be in order.  

[bkpragmatic]

BK School Near Me - bkpragmatic

BK School near Me: Where Learning Meets Innovation

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[ladyday3065]

The Artful Universe

In général, if there is some contextual association with an item of mathematics or music, then the language-specific thought-processes that deal with it should be awakened by contemplation of it. The relationship between music, pattern, and language invite us to map out some spéculative scénarios for their historical development. In the first instance, there exists some form of common ancestral mental function among humanity's forebears which split into separate threads -- one of music, the other of language -- while retaining some residual traces of the link between the two which is manifested in activities like singing.

In the second instance, music language is primary, and music subsequently evolves from it as a separate activity -- for example, because of the development of singing as a means of transmitting sounds over long distances.

Language could be a strand of human activity and culture that developed in parallel with a faculty for pattern- recognition.

At first, spacial pattern- recognition became well developed and blossomed into by-products, like art and image-making; then, temporal and pattern-recognition became acute, and diversified into musical rhythm. In this instance, music develops after other artistic practices. There might be a primary faculty for pattern-recognition, from which language-making split off. Therefore, temporal pattern-recognition developed into a further cultural branch in music,while the spatial pattern-recognition gave rise to art as a cultural manifestation. A primary pattern-recognition ability might gradually have diversified into more unique abilities: spatial pattern, temporal sequences, then language and numerical sequences. The 'meaning' of a mathematical formula is simply a finite claim of computation that have been used to construct it. This view may seem harmless enough, but it has dire consequences.

The constructivists wished to avoid manipulating counter-intuitive entities like infinite sets, about which we could have no concrete experience. As a result, constructivism grew to be known as intuitionism, in order to stress it's self-stated appeal to the foundation of human intuition.

A statement whose truth cannot be decided in a finite number of constructive steps remains in an undecided limbo status. Preconstructivist mathematicians, dating back with Euclid, had developed a variety of ways of proving formulas to be true that did not not correspond: a beloved of the ancient Greeks known as reducto ad absurdum. To show something as true, was assume it is false, and from that assumption deduce what must be contradictory (like 2=1). From that we make conclusions based on the facts left.

The theory of relativity: the multiple constituants of an experience, such an audible a mathematical equation made of temporal language processing, spatial imagery, and the waves on which sound travels in music, across history.

[pillmillipedes]

well, it doesn't have to work that way.

A good example would be mathematical constructivism (aka intuitionistic logic), a philosophical movement based on the assertion that to prove an object's existence, you must have an explicit way to construct it (contrasted with proof by contradiction).

It's characterized by not subscribing to the laws of excluded middle ("A ∨ ¬A always holds") and double negation elimination ("¬¬A implies A"), and consequently many everyday operations like De Morgan's laws, etc don't work in constructivism.

despite that, a good portion of classical analysis has been proven using it, so I could easily imagine a world where the some very influential mathematitian decided that proofs by contradiction aren't real proofs and got humanity stuck in this framework for a couple centuries.

Why are the laws of logic what they are, as opposed to something else?

This is a bit like asking "why are the laws of physics what they are?", but seemingly even less tractable.

A sufficient answer is not "the laws of logic were made up by man; ask him why he made them up that way!" or something of its ilk. It's true that all explicit systems of logic were devised by people, and that through history people have devised many different such systems in order to reason about the world. But it's likewise true that some conceivable systems of logical inference work better than others to that end. The inference rules of syllogistic logic, classical propositional logic, fuzzy logic, whatever, they all to one degree or another succeed at allowing one to derive new knowledge from old knowledge, new truths from known truths, etc. Some conceivable rules of inference, such as "from two statements, infer the first one" or "from A ∨ B infer ¬B" simply don't work for transforming old knowledge into new knowledge. They are unsuccessful methods of inference. And this would seem to say something about the underlying structure of the world, that some conceivable rules of inference work and others don't. We can, if we strain very hard perhaps, imagine that it was otherwise—we can imagine a world in which "from two statements, infer the first one" does work, although we can hardly say anything about what it would be like.

So, there's my question. Why does logic work the way it does, instead of some other way? Why is the world such that these rules of inference work and those rules of inference don't work?

I have no hope of ever getting an answer to this but I'm very curious.

[omegaphilosophia]

The Philosophy of Set Theory

The philosophy of set theory explores the foundational aspects of set theory, a branch of mathematical logic that deals with the concept of a "set," which is essentially a collection of distinct objects, considered as an object in its own right. Set theory forms the basis for much of modern mathematics and has significant implications for logic, philosophy, and the foundations of mathematics.

Key Concepts in the Philosophy of Set Theory:

Definition of Set Theory:

Basic Concepts: Set theory studies sets, which are collections of objects, called elements or members. These objects can be anything—numbers, symbols, other sets, etc. A set is usually denoted by curly brackets, such as {a, b, c}, where "a," "b," and "c" are elements of the set.

Types of Sets: Sets can be finite, with a limited number of elements, or infinite. They can also be empty (the empty set, denoted by ∅), or they can contain other sets as elements (e.g., {{a}, {b, c}}).

Philosophical Foundations:

Naive vs. Axiomatic Set Theory:

Naive Set Theory: In its original form, set theory was developed naively, where sets were treated intuitively without strict formalization. However, this led to paradoxes, such as Russell's paradox, where the set of all sets that do not contain themselves both must and must not contain itself.

Axiomatic Set Theory: In response to these paradoxes, mathematicians developed axiomatic set theory, notably the Zermelo-Fraenkel set theory (ZF) and Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). These formal systems use a set of axioms to avoid paradoxes and provide a rigorous foundation for set theory.

Set Theory and the Foundations of Mathematics:

Role in Mathematics: Set theory serves as the foundational framework for nearly all of modern mathematics. Concepts like numbers, functions, and spaces are all defined in terms of sets, making set theory the language in which most of mathematics is expressed.

Mathematical Platonism: The philosophy of set theory often intersects with debates in mathematical Platonism, which posits that mathematical objects, including sets, exist independently of human thought. Set theory, from this perspective, uncovers truths about a realm of abstract entities.

Philosophical Issues and Paradoxes:

Russell's Paradox: This paradox highlights the problems of naive set theory by considering the set of all sets that do not contain themselves. If such a set exists, it both must and must not contain itself, leading to a contradiction. This paradox motivated the development of axiomatic systems.

Continuum Hypothesis: One of the most famous problems in set theory is the Continuum Hypothesis, which concerns the possible sizes of infinite sets, particularly whether there is a set size between that of the integers and the real numbers. The hypothesis is independent of the ZFC axioms, meaning it can neither be proven nor disproven within this system.

Axioms of Set Theory:

Zermelo-Fraenkel Axioms (ZF): These axioms form the basis of modern set theory, providing a formal foundation that avoids the paradoxes of naive set theory. The axioms include principles like the Axiom of Extensionality (two sets are equal if they have the same elements) and the Axiom of Regularity (no set is a member of itself).

Axiom of Choice (AC): This controversial axiom asserts that for any set of non-empty sets, there exists a function (a choice function) that selects exactly one element from each set. While widely accepted, it has led to some counterintuitive results, like the Banach-Tarski Paradox, which shows that a sphere can be divided and reassembled into two identical spheres.

Infinity in Set Theory:

Finite vs. Infinite Sets: Set theory formally distinguishes between finite and infinite sets. The concept of infinity in set theory is rich and multifaceted, involving various sizes or "cardinalities" of infinite sets.

Cantor’s Theorem: Georg Cantor, the founder of set theory, demonstrated that not all infinities are equal. For example, the set of real numbers (the continuum) has a greater cardinality than the set of natural numbers, even though both are infinite.

Philosophical Debates:

Set-Theoretic Pluralism: Some philosophers advocate for pluralism in set theory, where multiple, possibly conflicting, set theories are considered valid. This contrasts with the traditional view that there is a single, correct set theory.

Constructivism vs. Platonism: In the philosophy of mathematics, constructivists argue that mathematical objects, including sets, only exist insofar as they can be explicitly constructed, while Platonists hold that sets exist independently of our knowledge or constructions.

Applications Beyond Mathematics:

Set Theory in Logic: Set theory is foundational not only to mathematics but also to formal logic, where it provides a framework for understanding and manipulating logical structures.

Philosophy of Language: In philosophy of language, set theory underlies the formal semantics of natural languages, helping to model meaning and reference in precise terms.

The philosophy of set theory is a rich field that explores the foundational principles underlying modern mathematics and logic. It engages with deep philosophical questions about the nature of mathematical objects, the concept of infinity, and the limits of formal systems. Through its rigorous structure, set theory not only provides the bedrock for much of mathematics but also offers insights into the nature of abstraction, existence, and truth in the mathematical realm.

[d6b-onion]

proship and anti-antiship being two distinct stances shows fanfiction discourse rejects the law of excluded middle

[philosophybits]

I shall try again and again to show that what is called a mathematical discovery had much better be called a mathematical invention.

Ludwig Wittgenstein, Lectures on the Foundations of Mathematics

[religion-is-a-mental-illness]

Non-believers often find themselves needing to have a comprehensive understanding of both religious scripture and the science that replaces it. One must not only know what Genesis says, and that it’s false, but also explain evolution and cosmology.

Believers, on the other hand, do not need to know even their own scripture, let alone hold any valid science literacy. “Faith” is sufficient.

This is similarly true of social theorists, where ignorance, scorn and outright denial of science, mathematics and statistics is not just acceptable amongst those pushing faith-based notions such as social constructivism and blank slateism, but an ideological moral and survival imperative.

When you’ve got a pretentious jargonese belief system to push, reality just gets in the way.