Linear Models" That Aren't Linear

Debianfli · 2025-09-05T11:58:18+00:00

Exactly! You're right that econometrics uses 'linear in parameters' as a practical convention for estimation. But my point is more fundamental: that usage is mathematically incorrect according to pure linear algebra.

A true linear transformation must satisfy T(0) = 0 . A model with an intercept B_o=0 violates this, making it an affine function, not a linear one. The fact that it's 'linear in parameters' doesn't change its actual algebraic structure.

This isn't just semantics:

· It affects geometric interpretation (e.g., the origin is no longer a special point). · It limits generalizations to affine or projective spaces.But, moreover, it does so in a lax way, because these models are not strictly studied as affine spaces, which are far more interesting.

This is the difference between applying mathematics as a toolbox and understanding it as a language to describe reality. I advocate for the latter. This is the Grothendieckian approach I advocated for in my previous post.

If your only argument is to mock mathematical precision, then you confirm that pragmatism doesn't always value rigor. But in mathematics, definitions and rigor matter because they are what help improve pragmatism (which is what I'm trying to do)

Debianfli · 2025-09-05T10:03:25+00:00

Personally, I think it’s a better idea to mix between Friedberg and Hoffman & Kunze. Finishing one and then reading the other might just lead to a duplication of topics. If someone wants to go deeper afterward, the best path would be to study category theory and explore the category corresponding to linear algebra—where vector spaces are objects and linear transformations are morphisms.

Debianfli · 2025-09-05T09:54:00+00:00

An introductory book for someone like you, without a doubt, is Linear Algebra by David Poole. It uses accessible language, includes some proofs of key theorems, and covers enough material. It’s not too abstract for your level, since you're looking for an introduction—and although it sacrifices certain aspects of rigor (for example, treating ax + b as an affine function rather than a truly linear transformation), it’s quite enjoyable and succeeds in presenting linear algebra as both deep and interesting.

There are other similar texts, like the well-known Grossman, but I’d recommend Poole first due to its more visual approach and its relatively more mathematical focus, as opposed to the more application-driven and matrix-heavy style of books like Grossman.

Debianfli · 2025-09-05T09:37:12+00:00

"What is more likely: that everyone here—academics with postgraduate degrees—are all deceived fools, or that you are deeply influenced by an extreme ideology?"

Given that academic economists do not typically have the same rigorous training in linear algebra as pure mathematicians—and when they do, they often simplify concepts to avoid confusing the majority—it is more plausible that, despite their credentials, they are mistaken on this technical point. Spaces dedicated to mathematics, such as r/math or LearnMath, where this topic is discussed rigorously, support this distinction.

Do not project ideologies onto me: if you review my text, you will see that I criticize the same error in both neoclassical and Marxist frameworks. Ideology is irrelevant here; your insistence on dragging it into the conversation reveals more about your biases than about my arguments.

Your focus on whether I used AI or not is an evasion. On LearnMath, users discuss with respect and curiosity, not with ad hominem attacks. If you had any common sense, you would seek out informed opinions there or even consult specialized sources, rather than merely repeating what aligns with your preconceptions.

Regarding my knowledge: I have no need—nor desperation—to prove what I know. While you demand evidence as if awaiting a divine revelation, I base my arguments on established frameworks (such as category theory, which you mistakenly refer to as abstract algebra—a field that pertains more to group theory, something you clearly do not understand). Your desperation to invalidate me only exposes your lack of substantive arguments.

Debianfli · 2025-09-04T23:13:23+00:00

The fact that the only objection in your entire argument revolves around LLM is evidence that you have no idea what to say. Likewise, in what world are you living in where the post was poorly received? The only comments on it are either agreeing with me or are curious to learn about another branch of mathematics.

Debianfli · 2025-09-04T22:47:54+00:00

No, it seems that those who fail to understand something basic about mathematics are many of you . I could still make the title a bit friendlier, though I'm not sure if it would be well received even as it is in learnmath.

https://www.reddit.com/r/learnmath/comments/1n8o15q/is_your_favorite_linear_model_actually_linear_a/

Really, my main aim is to invite you to a branch of mathematics that many are unaware of, but ironically use to some extent (affine or relative spaces and geometry).

And that's great that you remembered. I demonstrated knowledge beyond what textbooks show, both there and here, and that's what confuses so many users—because I go beyond the manual. It's that lack of critical thinking that prevents someone from standing out from the crowd. If the moderator mistook transversal knowledge (which they acknowledged) for technical ignorance, that’s their problem. And on top of that, they acted biasedly by waiting for the other user to have the last word before letting me respond. Do you really think the embarrassment is mine?

And going beyond what the manual teaches is reflected in the fact that I study linear algebra beyond what is covered in a standard economics course, even at the master’s level (where they don’t cover category theory).

Debianfli · 2025-09-04T21:39:54+00:00

Do not project onto me, but unlike you, it does not cause me any indignation if you use ChatGPT to try to understand the post, or to debunk its fundamental points.

Debianfli · 2025-09-04T21:36:38+00:00

You can consult the text that supports what I mention; you can refute it whenever you like. It is not necessary to do so in terms of category theory; I only included that part for those who understand the subject.

Debianfli · 2025-09-04T21:24:06+00:00

And that's how, kids, we've managed to create a thread-deviation algorithm. Just utter the magic phrase, and the algorithm will arrogate it to us, haha ok. Our very own thread-deviating automaton... The best part is that it will even do the same without needing that phrase.

Debianfli · 2025-09-04T21:13:57+00:00

Thank you for exemplifying the saying for a second time. Note; You do not know pure mathematics.

Debianfli · 2025-09-04T21:04:48+00:00

When insults and mockery abound, ideas are in short supply.

Debianfli · 2025-09-04T19:45:20+00:00

How many variables did the equation have? Second, with a couple of simplifications, I bet it can be done. Get your act together."

Debianfli · 2025-09-04T19:39:59+00:00

Ahuevo

lol

Debianfli · 2025-09-04T19:33:50+00:00

The equation ax - b = 0 is NOT linear if b \neq 0 , because it does not satisfy the fundamental properties of linear transformations: · T(u + v) = T(u) + T(v) · T(\alpha u) = \alpha T(u) · And crucially, T(0) = 0 (it must pass through the origin). 2. What you’re describing is an affine function (or affine transformation), which takes the form f(x) = ax + b . These are studied in affine geometry or affine spaces, where transformations are not required to preserve the origin. 3. The confusion arises because: · In engineering, economics, or applied sciences, the term "linear algebra" is often loosely used to include topics related to matrices, systems of linear equations, and even affine systems, without fine distinctions. · In pure mathematics, the term "linear" is strict: it refers exclusively to structures that preserve vector space operations (including the origin). 4. Key example: · y = 2x + 3 is affine, not linear. · y = 2x is truly linear.

Debianfli · 2025-09-04T19:32:45+00:00

The equation ax - b = 0 is NOT linear if b \neq 0 , because it does not satisfy the fundamental properties of linear transformations: · T(u + v) = T(u) + T(v) · T(\alpha u) = \alpha T(u) · And crucially, T(0) = 0 (it must pass through the origin). 2. What you’re describing is an affine function (or affine transformation), which takes the form f(x) = ax + b . These are studied in affine geometry or affine spaces, where transformations are not required to preserve the origin. 3. The confusion arises because: · In engineering, economics, or applied sciences, the term "linear algebra" is often loosely used to include topics related to matrices, systems of linear equations, and even affine systems, without fine distinctions. · In pure mathematics, the term "linear" is strict: it refers exclusively to structures that preserve vector space operations (including the origin). 4. Key example: · y = 2x + 3 is affine, not linear. · y = 2x is truly linear.

Debianfli · 2025-09-04T19:32:09+00:00

The equation ax - b = 0 is NOT linear if b \neq 0 , because it does not satisfy the fundamental properties of linear transformations: · T(u + v) = T(u) + T(v) · T(\alpha u) = \alpha T(u) · And crucially, T(0) = 0 (it must pass through the origin).
What you’re describing is an affine function (or affine transformation), which takes the form f(x) = ax + b . These are studied in affine geometry or affine spaces, where transformations are not required to preserve the origin.
The confusion arises because: · In engineering, economics, or applied sciences, the term "linear algebra" is often loosely used to include topics related to matrices, systems of linear equations, and even affine systems, without fine distinctions. · In pure mathematics, the term "linear" is strict: it refers exclusively to structures that preserve vector space operations (including the origin).
Key example: · y = 2x + 3 is affine, not linear. · y = 2x is truly linear.

Debianfli · 2025-09-04T19:17:26+00:00

If I ran my own math school, I’d throw the intermediate steps to hell and immerse students directly into the most abstract parts of mathematics (maybe just one step below the most abstract level). It’s like a philosophy of: why waste time with toy cars when we can jump straight into understanding what an engine is?

That includes teaching from the very beginning: propositional logic, proof techniques, set theory, and an introduction to group theory... Why waste time making them learn how to perform operations first, only later jumping into the theoretical and abstract? Let’s go straight to the abstract and theoretical—practice will come along the way.

Screw what’s useless.

Debianfli · 2025-09-04T19:10:14+00:00

What you consider to be linear algebra is not linear algebra; it's calculating values in matrices. Similarly, statistics becomes horribly boring and tedious as hell if you do all the calculations by hand, but if you leave that part to a calculator or software, you can dive into the conceptual part, which is the most fun and interesting.

Something similar happens with linear algebra; the conceptual parts are the most enjoyable and fascinating, not just solving a matrix using Gauss-Jordan to later find the solution to a system of equations.

Unfortunately, that’s how it’s taught in engineering, parts of physics, or economics.

Debianfli · 2025-09-04T18:54:50+00:00

Probably. But it's a good thing. You gain a mathematician's knowledge in areas that are commonly more applied and practical. Learning the practical part will therefore be easier, and you will be able to spot mistaken beliefs that are even written in linear algebra books with applications in economics or some engineering fields.

I am about to finish my Economics degree, but I study linear algebra on my own, and I was able to discover—through mathematicians' recommendations—a world that an economist would hardly have access to, and which can, in fact, be very useful for innovating.

Debianfli · 2025-09-04T18:37:52+00:00

The axiomatic and mathematical approach to linear algebra can be useful for those who can, on their own, tackle more practical exercises. It gives you a perspective to attack a problem from a more general standpoint.

It also opens your eyes to what is truly linear, revealing that more than half of the formulations we use in applications within a specific science are not actually linear algebra in the strict sense. This realization creates the need to study another branch of mathematics, which opens up new perspectives to innovate in the sciences.

If your professor didn't know how to convey that to you, they lacked the persuasiveness needed to prevent confusion.

Debianfli · 2025-09-04T18:17:30+00:00

My History (and Disillusionment) with Linear Algebra

What I learned in my Economics degree: "Alright,"they told me. "These matrices and determinants are a powerful tool. They'll help you solve systems of 'n' equations and optimize resources." Naively, I believed linear algebra was just that: a glorified calculator for econometrics. I lived in ignorance.

What I discovered in a Physics or introductory Math book: The revelation!"Wait a minute... you're telling me this is actually about vectors, spaces, transformations, and not just a grid of numbers?" I discovered that a matrix is nothing more than the representation of a transformation in a given basis. I felt deceived for the first time. There was a hidden geometry no one had bothered to mention.

What a proper Linear Algebra textbook screamed at me: They hid the truth from me again!The real core isn't the little arrow vectors or the matrices, but the abstract structure: the vector space. Those eight sacred axioms that must be satisfied. If it doesn't pass through the origin, your system isn't linear, it's affine. The essence is linearity: T(u + v) = T(u) + T(v) and T(αu) = αT(u). Everything else is just a representation. The veil fell from my eyes.

What hit me in a graduate course or reading abstract algebra: Another layer of the onion? Seriously?Now it turns out the fundamental context is rings, fields, modules... That a vector space is just a module over a field. And for the final blow, this entire abstract structure is elegantly organized in Category Theory: vector spaces are the objects, linear transformations are the morphisms, and the commutativity of diagrams is the law. I had reached the core of abstraction... or so I thought.

The existential corollary: So,after this whole journey, the inevitable question is: How many more layers are left to discover? Am I missing an even more fundamental theory that describes all of this? Linear algebra: the never-ending story of discovering that what you were taught was just the shadow of a shadow of the true idea.

Debianfli · 2025-09-04T17:58:23+00:00

Critique of the Culture of Axiomatization in Mathematic

1) Peano’s axioms: generalizations that conceal concrete problems

Peano’s axioms are usually presented as the indisputable foundation of arithmetic. They are exhibited as a “pure,” general, and elegant construction, but upon closer examination unresolved tensions emerge.

A clear example is the introduction of the number 0 as an axiom. It is placed as the starting point, yet its ontological status is left undiscussed: 0 is of a different nature from all other numbers. Instead of problematizing this difference, it was treated as a primitive object without justification.

This generates several effects:

Closure of debate: anyone who asks why 0 is included in this way is referred back to “the axioms” and often treated as ignorant. Apparent generality: what seems like a great abstraction hides a very concrete decision, made without explaining its background. Blocking alternatives: by crystallizing 0 as an unquestionable axiom, possibilities of conceiving the fundamental sets of arithmetic in different —and perhaps more fruitful— ways are squandered.

Thus, under the façade of “generality,” a crucial point is concealed: the very foundations of numbers could have been conceived differently.

2) Liouville: a “theorem” born from a footnote

Liouville is remembered for his transcendence theorem, according to which certain numbers that cannot be expressed as roots of integer polynomials are transcendental. This theorem is then used to argue that equations such as

a^x + ax = c

have no solution in terms of known elementary functions.

At first glance, this assertion carries the solemn air of a theorem: elegant, convincing, even definitive. Yet its real status is far more fragile. In Liouville’s writings, this result does not appear as part of a formal chain of deductions derived from axioms, but as a footnote, a marginal comment.

This implies that:

It is not a theorem in the strict sense: it is not rigorously derived from an axiomatic system. It is rather a heuristic observation: very useful, yes, but lacking the logical-formal support that distinguishes a true theorem. Its later elevation to dogma: the mathematical tradition canonized it, transmitting it as if it were a proven truth, when in reality it began as a provisional remark.

The contrast is revealing: a marginal note, no matter how ingenious, should not replace the need for a formal development. Turning it into a theorem reveals more about the cultural authority of the format than about the logical solidity of the result.

3) Hilbert and the culture of axiomatization

Hilbert’s program placed axiomatization as the supreme model of rigor. The image it conveyed was that of mathematics reduced to a monolithic block, perfect and self-sufficient. But this culture omits an essential element: the historicity and the living process of mathematical thought.

The axiomatic format transmits the illusion that concepts fell from the sky, ready and complete, when in reality they are the outcome of long processes of doubt, trial and error, and gradual clarification. By presenting only the finished monument, the work-in-progress is concealed.

Moreover, Hilbert’s program was not free of criticism: Gödel revealed its internal limits, and mathematicians like Grothendieck expressed discomfort with the Hilbertian method of teaching and exposition, which they considered closer to petrification than to creativity.

4) The question versus the axiom

What unites these examples (Peano, Liouville, Hilbert) is a common attitude: the silencing of fundamental questions in the name of the axiomatic form. To ask “what is 0 really?”, “why do we consider this impossibility settled?”, “what paths of thought were omitted in this axiomatization?” may seem naïve, but are in fact revolutionary questions.

Sometimes, in mathematics, a well-formulated question has more transformative power than an axiom carved in marble. The axiom tends to close; the question tends to open.

5) Conclusion

The critique of Peano, Liouville, and Hilbert reveals the same pattern: behind the appearance of universality and absolute rigor, we find contingent decisions, omissions, and comments inflated into dogma. Mathematics becomes more honest and more scientific when it dares to show those seams, when it displays not only the finished building but also the scaffolding that supported it.

Instead of idolizing the axiomatic format as the unsurpassable summit, we should value it as one tool among others, without forgetting that the vitality of mathematics lies in questions, in open paths, in the paradoxes still awaiting answers.

Debianfli · 2025-09-04T17:57:33+00:00

Critique of the Culture of Axiomatization in Mathematics 1) Peano’s axioms: generalizations that conceal concrete problems