You Don't Have to Rediscover the World: Learning from Other People's Thinking

Some of the hardest moments in learning come when you understand every word on the page. Each step seems reasonable. You can follow the argument while the book is open. Close it, though, and you are left with a handful of terms and no clear idea why anyone would have approached the problem that way. Change the exercise, and you need the book again.

It is easy to blame a weak foundation. You find another explanation, then another, and read the same definition in slightly different words. Sometimes that helps. Sometimes what is missing comes before the definition: a difficulty someone encountered, a limit to an older method, a reason this unfamiliar idea was worth introducing at all.

That is where I want to begin. When we study knowledge other people have already developed, we can borrow more than their conclusions. We can also borrow their reasons for arriving there. A textbook may leave much of that out, even when it is exactly what a beginner needs.

Where the textbook begins matters

“I understand” can mean several things. I can follow an example. I can explain the idea to someone else. I can tell whether a method still applies after the conditions change. These are different abilities, and different purposes call for different depths of understanding. Getting a particular job done does not always require working through the foundations of an entire subject.

One gap is especially worth noticing, though: knowing a rule without seeing why anyone would want that rule.

Definitions and formulas are finished work. To make knowledge easier to consult, use, and check, writers leave out much of the hesitation that preceded it. They reorder problems that were once tangled together. The result is clearer, but the way into it may no longer be visible. Something the author no longer needs to say may be the very step a reader is missing.

Consider a document describing how to use a software feature. This kind of specification tells you what information to supply, what result to expect, and what counts as an error. These details are essential when people need to use a system or make its parts work together. But if you do not yet know what job the feature is meant to do, even a precise specification can read like a list of unfamiliar terms. Watch it solve a problem first, and the same lines become easier to use.

I think of this editing and organization as a kind of compression, with one qualification: compression need not lose information. A rigorous statement may preserve all the relevant mathematical content while omitting the cues that would help a beginner approach it. Two accounts can express the same relationship and still demand very different amounts of work from the reader.

The difference becomes particularly noticeable when a concept initially seems arbitrary.

A number that seems to have no business existing

An introduction to imaginary numbers often begins with a rule: there is a number called i, and its square is −1.

The objection arrives almost immediately. Every real number we have learned about so far has a nonnegative square. How can this one be different? Looking for a physical counterpart makes matters harder. You can count one apple or cut half an apple. Where would you put i apples?

Leave that question open for a moment and consider a sixteenth-century problem:

x³ = 15x + 4

The question asks us to take three copies of a number and multiply them together. The result must equal fifteen times that number, plus four. Try 4, and both sides give 64. Yet applying the available cubic formula to this equation produces the square root of −121 along the way.¹

The answer is there, but one route to it seems impassable. For someone trying to make sense of the formula, that is harder to dismiss than a problem with no answer at all.

In his Algebra, published in 1572, Rafael Bombelli treated calculations with such quantities systematically. Using modern notation, we start with i² = −1, which makes 11i a square root of −121. The formula then asks us to find two numbers whose cubes are 2 + 11i and 2 − 11i, and add those numbers together.

Multiplying under the new rule gives:

(2 + i)³ = 2 + 11i

(2 − i)³ = 2 − 11i

We can therefore choose 2 + i and 2 − i as a compatible pair of the numbers the formula requires. Add them together, and the imaginary parts cancel, leaving 4.

Even without working through the multiplication yourself, you can now see the difficulty. A problem with an ordinary answer asks us to make room for an unfamiliar kind of number during the calculation. With suitable rules, a calculation that seemed to stop can continue.

One successful example does not prove an entire mathematical system, or answer everything we might ask about imaginary numbers. It answers an earlier question: why would anyone take such a thing seriously? The operations, proofs, and geometric interpretations that follow now have something to build on.

The order matters to me. An abstract concept need not correspond to a tangible object before it can begin to make sense. Sometimes understanding starts with seeing which relationships it preserves and what it allows us to do.

Precise definitions have problems to solve, too

Seeing a method work still leaves something to explain. One problem calculus addresses is how to find an object’s speed at a particular instant. Dividing the distance traveled over a stretch of time by the time taken gives us an average speed. But however short the interval, an interval is not an instant.

Early methods for dealing with change produced substantial results, but the reasoning behind them also drew criticism. In The Analyst, published in 1734, George Berkeley pressed a question about vanishing increments in certain derivations: how could a quantity be used as a divisor and then be allowed to disappear?²

You need not have studied a full calculus course to see the concern. If a quantity is zero, you cannot divide by it. If it is not zero, its smallness alone does not make it nonexistent. Useful answers still leave a need to explain why the reasoning holds.

The modern idea of a limit gives us a more precise approach. In the speed example, the time interval remains nonzero in every calculation. We ask whether the average speed approaches a definite value as the interval gets shorter. This is different from dividing by zero. A beginner who encounters only the nested conditions in a formal definition may wonder why mathematicians have made things so difficult. Knowing which ambiguity those conditions address gives them a purpose.

Different periods had different ways of justifying results, and later foundational work was not completed in a single step. A story of intuition simply giving way to rigor would miss much of this history. I want to borrow a more limited turning point for learning: after a method has produced results, there is still work to do in examining which operations it permits.

Approaching a definition through its use does not reduce the demand for precision. When I can see what goes wrong without a condition, I have a better reason to remember it than the possibility that it might appear on an exam.

Understanding someone who does not yet know the answer

Empathy usually brings feelings to mind: why someone is hurt or angry, or why a seemingly minor event matters so much to them. Thinking has its circumstances, too. An approach can look needlessly roundabout because we already know the later solution and have forgotten which tools were unavailable at the time.

Education researchers use the term epistemic empathy for understanding and appreciating someone’s cognitive and emotional experience while constructing, communicating, or evaluating knowledge. In a study by Jaber and colleagues, the setting was teacher education: helping teachers understand learners.³

I would like to turn that direction around as an approach to independent learning. Can a reader try to understand the situation faced by the person who developed an idea? This is my extension of the approach, not a result established by that study.

It does not require writing imaginary inner monologues for people who died centuries ago. More reliable questions concern the surviving evidence. What problem was the person working on? What methods were available? Which difficulty did a particular choice resolve, and which new difficulties did it create? Where the record is silent, we should leave a gap. Where we rearrange events for teaching, we should call it a reconstruction.

With Bombelli, we do not need to guess how astonished he felt on encountering a negative square root. Looking carefully at the problem, the formula, and the calculation is enough to show why the attempt was worth making. History gives us a place from which the eventual answer does not yet seem obvious.

The same habit can help outside mathematics. When reading a philosophical argument, identifying the position the author opposed may do more for understanding than memorizing a celebrated sentence. A statement has a different force depending on the question it answers. Remove the disagreement and leave only the elegant conclusion, and we may assign the author a problem they never intended to address.

Reading widely can leave us with more than facts

If learning includes this ability to change perspective, breadth has another use. Exposure to different fields lets us see how people decide what is worth asking about.

Programming directs attention to how a task can be divided and what its parts must agree on. Statistics asks us to distinguish an observed difference from an inference the evidence can support. History encourages us to examine where a statement comes from and the circumstances in which it was made. These habits can travel, although reading a book does not automatically make someone good at applying them elsewhere.

Suppose a piece of work repeatedly arrives late. A systems perspective suggests checking the handoff: does work one person considers finished still need to be redone by the next? Another approach asks what “repeatedly” means. How many cases have there been? Has the recent work been harder? No discipline can establish the cause by analogy alone. These approaches offer different questions that can actually be investigated.

Borrowed approaches can also mislead. People cannot be treated entirely as software components, and a few observations do not amount to sufficient data. Crossing into another field requires checking which relationships still hold. Making a connection is a beginning; noticing where the analogy fails helps make the borrowing dependable.

For this reason, I would not describe the idea as “empathy raises IQ.” That is a separate psychological claim requiring evidence. What I mean is more specific: understanding someone else’s intellectual situation can help me learn questions I would not otherwise have thought to ask. It gives me a few more approaches to try when the next problem arrives.

How far back is far enough?

By now, there is an obvious practical objection: time. If imaginary numbers require a detour through sixteenth-century algebra, and calculus requires centuries of foundational debate, when do we get to do anything?

Tracing ideas to their origins should not become another obligation attached to learning. Often it makes sense to accept the finished result. You can cook a meal from a recipe without first studying the chemistry of every ingredient. You can get a program running from an example before understanding all its design choices. Learning depends on being able to use knowledge that other people have already organized.

Looking back becomes worthwhile when the same difficulty keeps stopping you. A small change to a problem leaves you unsure what to do. You remember a rule but cannot tell where it applies. Three explanations later, a definition still seems like an arbitrary decree. Another similar summary may not fill that gap.

I would first look for one concrete problem the concept was meant to solve. If its purpose is still unclear, I would examine the limitations of an older method. The aim is to go back only far enough to reconnect with the learning at hand. Understanding one page should not require finishing an entire intellectual history.

A reasonable stopping point is modest. Can I explain what the idea helps with, use it once under its current definition, and identify a situation where I cannot simply carry it over? If so, I can move on for now. A later difficulty may give me a reason to return.

History is untidy, and it is not automatically a good syllabus. Old notation can be hard to read; some disputes contribute little to the immediate question. Good teaching can give us an answer that took others years to find while preserving one or two necessary turns in the reasoning. Saving learners unproductive searching leaves them more time to examine the steps that matter.

Ask AI for more than another explanation

Search engines can help locate these missing pieces, and AI can help organize them. But “explain this simply” may produce a friendlier account with exactly the same gap.

A more specific request would look like this:

I am learning ___ and can follow the examples, but I do not understand why this concept is needed. Start with a problem I can understand. Show how I could approach it without the concept and where that approach runs into limitations. Then introduce the concept and return to its current standard definition. Explain any prerequisite knowledge the example requires. For historical claims, provide sources I can check and distinguish documented history from a reconstruction designed for teaching. Do not turn missing information into a story. Finish with a problem in which one condition has changed, and withhold the answer for now.

This asks for help arranging a way into the problem. Dates, names, and quotations still need checking, as does the reasoning. A smooth account can make it easy to forget that we have only heard a version that sounds plausible.

It is also worth stopping to attempt that final problem. When following someone else’s reasoning, each step supplies a cue for the next. Working alone reveals the connections you cannot yet make.

Take i from the earlier example. If you have learned to multiply expressions in parentheses, try a related calculation: expand (3 + i)² and account for each term. If your immediate purpose is conceptual, try explaining in your own words why “we cannot find it among the real numbers” is not, by itself, enough to rule out extending the number system. Then investigate how the extension preserves the original operations. Different goals call for different exercises, but each should include some work that copying the text cannot do for you.

One further distinction matters. Understanding why someone proposed a claim does not establish that the claim is correct. Mathematics still requires proof; claims about the world still require evidence. Even when studying established knowledge, we need to distinguish proven results from approximations and unsettled questions. Understanding a person’s circumstances can help us read them. It cannot settle whether we should agree.

Coming back with something we can use

Part of what I value in this approach is that it gives difficulty another possible explanation. We may lack the background, or we may not have spent enough time. But the author may also be standing beyond a solved problem while the reader has not yet seen the problem at all. Recovering a little of its origin can help more than adding another set of terms.

In No View Is the Whole, I use retention, combination, and return to examine how understanding develops and revises itself. In independent learning, these can be ordinary actions: accept a usable account, connect a new concept with what you already know, and return to examine what was left out when the connection fails. Sometimes the content needs to change. Sometimes we need a different order of approach.

The formulas, proofs, writing, and diagrams people leave behind have already saved later generations years of work. We can borrow a little more by asking which problems those results once answered. An unfamiliar rule gains a reason. Then we can take it back to our own work and find out how far it applies.

The next time a definition refuses to make sense, try writing a question beside it: “Without this, where would I run into trouble?” Find one sufficiently clear example, then return to the page. Sometimes we do not need another explanation so much as somewhere for the existing one to begin.

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Further reading: No View Is the Whole: A Second-Order Philosophy of Understanding, Action, and Revision. This essay stands on its own; no prior knowledge of the book is needed.

1. For Bombelli’s work, the publication of his Algebra in 1572, and his treatment of complex arithmetic, see J. J. O’Connor and E. F. Robertson’s MacTutor biography. The calculation uses modern notation; see also Heinz Klaus Strick’s essay on Bombelli (PDF). The example selects compatible cube roots rather than treating complex cube roots as single-valued. The product of 2 + i and 2 − i is 5, satisfying the formula’s pairing condition for this equation. ↩︎

2. George Berkeley, The Analyst (1734), particularly sections XIII–XVI on the elimination of increments. Full text edited by David R. Wilkins, Trinity College Dublin. The explanation here uses modern language; it neither adopts all of Berkeley’s reasoning nor suggests that his criticism invalidated calculus. ↩︎

3. Lama Z. Jaber, Sherry Southerland, and Felisha Dake, “Cultivating epistemic empathy in preservice teacher education,” Teaching and Teacher Education 72 (2018), 13–23. DOI: 10.1016/j.tate.2018.02.009. The study concerned teacher education. The learner-to-thinker direction proposed here is an extension, not an outcome directly tested in that research. ↩︎