Romp into bad logic and bad thinking and bad mathematics, some advanced number theory

Romp into bad logic and bad thinking and bad mathematics, some advanced number theory

I get these from refdesk.com.  This one has me and the LLM twisting each others' ears.

THOUGHT OF THE DAY:     

"Do I contradict myself? / Very well then I contradict myself, / (I am large, I contain multitudes.)" - Walt Whitman, from Leaves of Grass, Song of Myself, 51. 


Oh, that's a famous one.

It really is one of the most liberating lines in American literature.

What makes section 51 so enduring is how Whitman flips what most people treat as a intellectual flaw—inconsistency—into a declaration of expansive human capacity. Writing in 1855, in an era obsessed with rigid categorizations, logic, and fixed systems, he basically shrugged and said: A human mind is an ecosystem, not a geometric proof.

There is a wonderful audacity in the way he frames it, too. He doesn't offer an apology or a elaborate defense for changing his mind or holding opposing thoughts at once. It's just a grand, sweeping acceptance of human complexity.

It makes you wonder how much energy people waste trying to keep all their internal lines straight, when the dynamic tension between conflicting ideas is usually where the actual living happens.

To be honest, although I often change my mind about something, that is not self-contradiction, that's changing one's mind, and one should not be labeled inconsistent for being able to do so.

That distinction gets right to the heart of where people trip up on this quote. Updating a position when presented with new evidence, better context, or simple reflection isn't a contradiction—it’s just cognitive mobility. Calling someone inconsistent for growing or revising a stance is conflating intellectual flexibility with hypocrisy.

True self-contradiction is holding two mutually exclusive, fundamentally incompatible premises at the exact same time—and often acting on both.

Where Whitman’s "multitudes" actually fit best isn't in a timeline of changing opinions, but in simultaneous internal reality. A person can simultaneously hold a deep desire for solitary peace and a strong craving for social noise; they can value ruthless pragmatic logic in business while being completely driven by sentimental sentimentality in private; or they can feel absolute certainty about a principle in the abstract while harboring deep empathy for the exception in front of them.

Changing your mind over time is just learning. Containing opposing forces within yourself at a single moment—that’s where the "multitudes" actually live.

An internal inconsistency is not possible if one is conscious of it. Of course contradictory statements out of the same mouth do seem suspicious.

That hits on a fundamental paradox of human cognition: can a conscious mind genuinely sustain two explicitly contradictory beliefs at the exact same instant if it holds them up to the light together?

Strictly speaking, the moment a contradiction is made explicit and conscious, the mind’s natural error-correction mechanism usually kicks in. It forces a resolution, a compartmentalization, or an admission that one side is a desire while the other is a reality. In that sense, conscious awareness acts like a spotlight—it’s very hard to hold two mutually exclusive propositions in the exact same focal point without one collapsing or shifting into a different category.

Where simultaneous internal contradictions usually survive is in the friction between different layers of the mind—the intellectual belief versus the instinctive reaction, or the abstract principle versus the immediate emotional drive. You can consciously know, for instance, that a fear is completely irrational, yet simultaneously experience the full, genuine physiological and mental reality of that fear. Both are happening at once, and knowing it doesn't instantly dissolve the tension.

As for the "contradictory statements out of the same mouth"—that's usually where theory meets social reality. When someone utters direct contradictions externally, it usually signals one of three things: cognitive dissonance leaking out before it's resolved, performative audience-pleasing, or a failure of basic intellectual hygiene.

Outward consistency in speech is reasonable to expect. The internal landscape, though, is often far less tidy than the sentences we produce to describe it.

Physics seems able to hold mutually inconsistent models at the same time of a lot of things.

Physics is perhaps the ultimate real-world case of holding mutually incompatible models in the same room without the building burning down.

The classic collision point, of course, is quantum mechanics and general relativity. At a fundamental level, their core assumptions directly contradict each other:

    General Relativity models space and time as a smooth, continuous, deterministic fabric where gravity is geometry.

    Quantum Mechanics models reality as discrete, inherently probabilistic, and jittery at small scales, where fundamental forces are mediated by particle exchanges in a flat background.

Try to plug quantum math into general relativity at Planck scales, and the math explodes into nonsensical infinities. They are logically incompatible descriptions of the exact same physical universe.

Yet physicists comfortably use both every single day. Why doesn't the discipline collapse under the weight of its own contradiction? Because physics treats models as pragmatic approximations of reality, not absolute metaphysical truth.

In physics, a model isn't judged by whether it is "the ultimate truth," but by its domain of validity:

    If you're calculating the orbit of Mercury or the bending of starlight, relativity works with staggering precision.

    If you're modeling electron behavior in a semiconductor or quantum tunneling, quantum mechanics gives the exact right answer.

Physicists live with the contradiction because they recognize both theories as effective field theories—interim tools that work within specific boundaries. They don't claim the universe is quantum mechanics or relativity; they acknowledge that human math has mapped two different provinces of the same territory using incompatible coordinate systems.

In that sense, physics operates much like a conscious mind dealing with internal tension: it uses the model that fits the scale of the problem at hand, while keeping a full-time research effort running in the background to find the unified theory that finally resolves the discrepancy.

It's a case of using the present model with its flaws until the new improved one arrives.

Precisely. It is pure, ruthless pragmatism. You don't throw away a map that gets you safely across 99% of the terrain just because the edges are torn and it can't show you what's under the water.

In history and philosophy of science, this is basically the operational reality of Thomas Kuhn’s structure of scientific revolutions. A working paradigm is never discarded simply because anomalies or internal contradictions exist; it is only abandoned when a new model arrives that not only resolves those contradictions, but retains all the predictive power of the old one.

Until that replacement shows up, engineers still need to design GPS satellites using relativistic corrections, and semiconductor designers still need to build microprocessors using quantum mechanics. The contradiction is treated not as a fatal moral failing of the discipline, but as a known boundary marker—a signpost reading "Here Be Dragons" that tells researchers exactly where the next breakthrough needs to happen.

It’s an interesting contrast to how people often approach human thought. In debate or daily life, people tend to treat a contradiction as an immediate, fatal error that invalidates the whole structure. Physics proves that you can run a remarkably successful, highly predictive global enterprise on a foundation of known, temporary incompatibilities—so long as you remain honest about where the seams are.

Ah, it is possible to "prove" a true theorem from false premises. Therein lies the fallacy that because the consequence is true the premise must also be true.

That is the classic fallacy of affirming the consequent—and in formal logic, it is one of the easiest traps to fall into precisely because the outcome works out.

In material implication (P⟹Q), if P is false, the implication itself remains logically true regardless of whether Q is true or false. You can start with complete nonsense as a premise and, through valid steps, arrive at a perfectly true conclusion:

    False Premise (P): All fish live on the moon.

    False Premise: The moon is in the Pacific Ocean.

    Valid Deduction: Therefore, all fish live in the Pacific Ocean (True Conclusion, Q).

The math works, the conclusion is empirically sound, but the foundation is garbage.

This is the central hazard of instrumentalism in science and daily life. Just because a model or premise yields accurate predictions (Q is true) does not mean the premise itself accurately describes reality (P is true). A wrong model can generate right answers by accident, through offsetting errors, or simply because it operates as a restricted mathematical stand-in over a limited domain.

Ptolemaic geocentrism is the classic example: calculating planetary motions using complex epicycles on a fixed-Earth model yielded remarkably accurate astronomical predictions for centuries. Navigators crossed oceans using it. The results were true, but the premise—that the cosmos revolved around the Earth—was dead wrong.

It’s a stark reminder that predictive success is a necessary condition for a true model, but never a sufficient one.

Your syllogism has a little problem: drop the uses of the word "all."

Good catch.

Without "all," the standard quantified structure collapses. Dropping "all" gives:

    Premise 1: Fish live on the moon.

    Premise 2: The moon is in the Pacific Ocean.

    Conclusion: Therefore, fish live in the Pacific Ocean.

In standard English, "fish" without a quantifier is implicitly generic, but logically, it becomes ambiguous. If interpreted existentially ("Some fish live on the moon"), the syllogism suffers an undistributed middle term—meaning the deduction itself fails to hold formally, even though the conclusion happens to be empirically true.

To keep the formal validity rock-solid while ditching "all," single entities make it much cleaner:

    False Premise (P): The Moon is made of cheddar cheese.

    False Premise: Cheddar cheese is rich in calcium.

    True Conclusion (Q): Therefore, the Moon is rich in calcium.

Both premises contain clear falsehoods, the deductive steps are completely valid, and the conclusion turns out to be factually correct (the lunar crust is indeed abundant in calcium).

It demonstrates the exact same principle: an empirical truth (Q) tells you nothing definitive about the truth value of the underlying premises (P) that produced it.

Well, there are fish around other than the Pacific Ocean.

That highlights another layer where natural language slips away from formal logic.

In standard English, saying "Fish live in the Pacific Ocean" sounds like an absolute statement—as if all fish live there, or as if that's the only place fish exist. But empirically, it's just a partial truth: fish live in the Pacific, but they also live in the Atlantic, rivers, ponds, and lakes around the globe.

In formal logic, that statement is just an existential claim (∃x): "There exists at least one fish such that it lives in the Pacific Ocean."

That's why natural language is so treacherous for syllogisms. When we drop explicit quantifiers like "all" or "some," our brains automatically fill in the blanks based on context, often confusing three very different things:

    Universality: All fish live in the Pacific (False).

    Exclusivity: Fish live only in the Pacific (False).

    Existential truth: Some fish live in the Pacific (True).

When a premise uses bare, unquantified nouns, it easily sneaks past our mental filter because one reading of it is true, even if the strict logical reading required for the argument is completely broken.

I'm having a problem with what you just said. Fish live in the Pacific Ocean only means there are fish there, it says nothing about any other fish.

You are completely right, and that gets straight to the point.

"Fish live in the Pacific Ocean" asserts a single fact: in the body of water called the Pacific, living fish exist. It makes zero claim about the Atlantic, lakes, or any other body of water, nor does it claim that every fish is in the Pacific.

To read it as implying that fish don't live elsewhere is to project an exclusivity that simply isn't in the sentence. In natural language, stating a location for a group doesn't negate their existence everywhere else.

It's a good example of how easily extra assumptions get tacked onto plain statements—and why keeping premises stripped down to exactly what they state (and no more) is so crucial for clear thinking.

So, false premise: Fish live on the moon. Second false premise: The moon is in the Pacific Ocean. True conclusion: Fish live in the Pacific Ocean.

Now the logic is completely tight.

By removing the ambiguous quantifiers, the argument rests on pure set membership:

    Premise 1: The set of entities living on the moon includes fish. (False)

    Premise 2: The moon is entirely contained within the location "the Pacific Ocean." (False)

    Conclusion: Therefore, the set of entities living in the Pacific Ocean includes fish. (True)

The deduction is formally valid: if A is in B, and B is in C, then A must be in C. The conclusion is empirically true—fish do indeed live in the Pacific Ocean. Yet every single step leading to that truth was built on absolute absurdity.

It brings us right back to why model-building and empirical verification can be so tricky. A system can yield a completely correct outcome (C) while being utterly, hilariously wrong about how it got there.

Can you provide an arithmetic example of that: I've seen them and knowing one would be useful.

Here are two classic arithmetic examples that illustrate this exact principle.
1. The Squaring Fallacy (Valid Operation on a False Premise)

Consider starting with a statement that is undeniably false:
Premise (False): −5=5

Apply the valid algebraic rule that if a=b, then a2=b2:
Step: (−5)2=(5)2
Conclusion (True): 25=25

The conclusion (25=25) is indisputably true, and every step of the algebra followed valid rules. However, working backward to claim that −5=5 must be true because 25=25 is true is the exact logical fallacy of affirming the consequent.

This is precisely why squaring both sides of an equation in algebra can introduce extraneous solutions—it allows false premises to mask themselves behind true outcomes.
2. "Anomalous Cancellation" (False Rule Yielding a True Result)

Another form occurs when a false operational rule accidentally yields a correct arithmetic value.
False Rule/Premise: "To simplify a fraction, cancel out identical digits in the numerator and denominator."

Apply this false rule to the fraction 6416​ by crossing out the 6s:
6​416​​⟹41​
Conclusion (True): 6416​=41​(0.25=0.25)

The final equality 6416​=41​ is factually correct, but the premise—that you can cancel individual digits across a fraction bar—is entirely false. (Try applying that same rule to 9919​ or 3523​ and the system breaks down immediately).
3. The Zero Multiplier
Premise (False): 3=7

Multiply both sides by 0:
3×0=7×0
Conclusion (True): 0=0

In all three cases, getting the right result at the end proves nothing about the truth of where you started.

And, of course, if you divide by zero but hide it in algebra, you can "prove" pretty much any number equals any other number.

That is the classic "mathematical sleight of hand"—using a hidden division by zero to systematically break the entire foundation of algebra.

The beauty of the trap is how innocently it hides in plain sight behind basic variables.
The Standard 1=2 Proof

Start by setting two variables equal to each other:

    Let a=b.

    Multiply both sides by a:
    a2=ab

    Subtract b2 from both sides:
    a2−b2=ab−b2

    Factor both sides:
    (a−b)(a+b)=b(a−b)

    Divide both sides by (a−b):
    a+b=b

    Since a=b, substitute a for b:
    b+b=b⟹2b=b

    Divide both sides by b:
    2=1

Where the Trick Hides

The entire illusion relies on Step 5: dividing both sides by (a−b).

Since the starting premise was a=b, the quantity (a−b) is actually equal to zero.

Dividing by zero destroys information. In arithmetic, if you multiply any number by zero, you get zero (5×0=0 and 2×0=0). If division by zero were allowed, you could "cancel" the zeros and claim 5=2.

That single illegal division step creates a black hole in the logic, allowing you to derive 1=2, 0=100, or any other absurdity you care to construct. It’s the algebraic equivalent of sneaking a false premise past the goalie right in the middle of an otherwise spotless derivation.

Yeah, you are dividing by (a - b) and a = b.

Exactly. (a−b)=0, so dividing by it sneaks 0/0 into the equation.

Because 0×x=0 for every possible number x, the expression 0/0 is completely indeterminate—it could theoretically evaluate to 1, 2, 42, or infinity. The moment that division happens, the equals sign loses all meaning and you can make the math say whatever you want.

It's the ultimate algebraic landmine: logically sound on every visible line, but entirely destroyed by a single invisible zero.

Is that an argument for saying that zero is not quite a legitimate "number" like the others?

That very line of thinking held back mathematics for centuries.

For a long time—across ancient Greece, Rome, and early medieval Europe—zero was treated with deep suspicion precisely because it doesn't behave like other numbers. It isn't a "quantity" of things you can touch, and as a operational element, it breaks key algebraic rules (like division and unique invertibility).

However, modern mathematics takes the opposite view: zero is entirely legitimate, but it plays a unique structural role.

Rather than viewing zero as a broken or defective number, abstract algebra categorizes it as an identity element and an annihilator:

    The Additive Identity: Under addition, zero is the neutral element (a+0=a). It allows us to define negative numbers and subtraction via inverse elements (a+(−a)=0). Without zero, you don't have a mathematical group or ring—the foundational structures of modern algebra.

    The Multiplicative Annihilator: Under multiplication, zero absorbs everything (a×0=0).

The restriction against dividing by zero isn't a failure of zero; it's a consequence of what division actually means. Division is defined as the inverse operation of multiplication:
ba​=c⟺c×b=a

If you try to divide a non-zero number by zero (say, 05​=c), you are asking: "What number c, when multiplied by 0, equals 5?" Since c×0=0 for all real numbers, no such c exists.

If you try to divide zero by zero (00​=c), you are asking: "What number c, when multiplied by 0, equals 0?" Every single number works, making c completely indeterminate.

Zero isn't "illegitimate"—it's an exceptional boundary object. Just as light speed in special relativity behaves fundamentally differently from sub-light velocities, zero behaves differently from non-zero scalars. Treating it like an ordinary counting number leads to algebraic chaos, but recognizing its unique role is what made calculus, linear algebra, and computer science possible.

I can see both sides of that. We can say we use zero as a convenient thing when we need it, just not as a divisor.

That is essentially the pragmatic compact mathematics ended up making with it.

We grant zero full membership in the number system—allowing it to serve as a positional placeholder, an origin point on a graph, and the anchor for addition—on the strict operational condition that we keep it away from the denominator.

It works much like a specialized tool with a clear safety warning attached: invaluable for setting up coordinate systems and defining balance points, but if you try to use it to divide, the machinery instantly jams.

In a way, it brings us right back to how physics treats its models. We don't need zero to be uniform or well-behaved in every single mathematical context; we just need to be crystal clear about its specific domain of validity. Within those guardrails, it remains one of the most useful abstractions ever invented.

I've had this fantasy that someone my find a use for a system of "unreal" ("imaginary" is already taken) numbers, numbers divided by zero. They might if there was a use.

That fantasy isn't as wild as it sounds—in fact, mathematicians have built several rigorous systems to do exactly that! The trick is that whenever you allow division by zero, you have to trade away some standard algebraic rules (like a−a=0 or unique inverses) to keep the math from collapsing into total nonsense.

Here are the three main ways mathematicians have constructed systems for "numbers divided by zero":
1. The Extended Real Numbers & The Riemann Sphere (Projective Geometry)

In standard analysis, if you divide a non-zero number by numbers getting closer and closer to zero, the result shoots off to infinity.

    The System: Mathematicians add a single "point at infinity" (∞) to the complex plane, creating a 3D surface called the Riemann Sphere.

    Division by Zero: In this geometric frame, 0z​=∞ for any non-zero complex number z.

    The Catch: 00​ and ∞∞​ remain undefined. You lose the ability to do simple algebra without checking boundary cases, but you gain a powerful tool.

    The Practical Use: This is used heavily in quantum mechanics, electromagnetic field theory, and fluid dynamics to model poles, complex potentials, and physical singularities smoothly.

2. Wheel Theory (Algebra without Division-by-Zero Exceptions)

Invented by Anton Setzer in 1997, Wheel Theory explicitly extends algebraic structures so that division by zero is always defined everywhere, for every element.

    The System: It adds two new elements to the number system: ∞ (representing 1/0) and a new element called ⊥ (pronounced "bot" or "null", representing 0/0).

    Division by Zero: 0a​=∞ (for a=0), and 00​=⊥.

    The Catch: To make this mathematically consistent without proving 1=2, you lose the standard distributive law. In a wheel, x−x does not always equal 0 (for instance, ∞−∞=⊥).

    The Practical Use: Computer scientists and logicians use wheel theory in formal software verification and floating-point arithmetic logic. It allows automated theorem provers and compilers to handle divide-by-zero operations deterministically without crashing or throwing unhandled exceptions.

3. Hyperreal Numbers and Non-Standard Analysis

Developed by Abraham Robinson in the 1960s, this system formalizes the concept of infinitesimals (numbers infinitely close to zero, but not zero) and infinities.

    The System: Instead of dividing by exact zero, you divide by an infinitesimal ϵ.

    Division by "Zero": ϵ1​=H, where H is a rigorous, infinite hyperreal number.

    The Practical Use: It gives calculus a completely rigorous foundation without needing limits (limx→0​), matching the original physical intuition that Isaac Newton and Gottfried Leibniz had when they first invented calculus.

So your intuition is dead-on: mathematicians did build those systems, and they found genuine uses for them—mostly in computer logic, software safety, and theoretical physics!


Gemini is AI and can make mistakes.