Problem-Solving --- From Blurry Thinking to Focused Thought

1. Introduction

"A sudden inspiration is often the result of a long period of hard work." – Thomas Edison

A central but often overlooked feature of problem-solving is that it does not begin with clarity. Contrary to the polished presentation of solutions in textbooks and formal instruction, real thinking typically starts in a state of vagueness. The problem is only partially understood, relationships are indistinct, and ideas exist as fragments rather than structured arguments. This condition, what may be called blurry thinking, is not a defect of reasoning but its natural starting point (Hadamard, 1945).

The work of mathematician Jacques Hadamard provides one of the most influential accounts of how such indistinct thinking becomes precise. Building on insights from Henri Poincaré[1] and Graham Wallas (an English sociologist), Hadamard described problem solving as a staged progression through preparation, incubation, illumination, and verification (§3-§6). This framework captures the essential movement from cognitive blur to focused, structured thought. In fact, Hadamard gives full credit to Wallas for the ideas he pursued.

This chapter presents a different problem-solving model, where brainstorming is unhelpful and solutions are largely clear, needing only diligent effort.  It is for very deep problems where one begins with no “clue” whatsoever. Lawyers may experience this in difficult cases, and, of course, scientists do. Even knowing where to begin can be a problem.

What Edison implies in his quote is that inspiration doesn’t always come as a well-formed sentence. It arrives in the language of the brain, not yet a subject taught in language departments. Yet, Hadamard, Poincaré, Edison, and all who think for a living would agree that hard work is part of this equation. In §8, we consider specific examples, modern and historic.

You might say this chapter is something like a graduate class on problem-solving.

2. Blurry Thinking is the Starting Point

Problem-solving begins in a diffuse cognitive state. Of course, there is some problem in mind, and the mind has been placed wide open for anything. The thinker lacks a clear representation of the problem and must work with incomplete, ambiguous, or even contradictory elements. Ideas often arrive non-verbally, appearing as images, partial patterns, or intuitive impressions. That is, blurry. Importantly, the blurry phase should not be avoided or prematurely suppressed.

This stage is frequently misunderstood, if understood at all. Because it lacks clarity, it may be mistaken for confusion or incompetence. In reality, it is a necessary condition for deep thinking. Without this initial openness, exploration is constrained, and genuine discovery becomes unlikely. The mind has internalized “chunks” of theories and how they work or interact with other chunks[2]. Call this chunk theory, if you like.

This may seem a very imprecise way to solve big problems, given that we are trained from the early grades to think with words, articulating steps clearly and linearly. However, most of us have received – almost as if by radio signal – a burst of insight from which we quickly try to articulate what it was. Yet, sometimes, often I’d say, it disappears as quickly, and we struggle to recapture what it was. What Hadamard attempts to teach us is to allow the blurry, even fuzzy, thought processes to further develop that burst. At the risk of a poor analogy, we’ve all had such bursts, say when needing to go to the kitchen for some reason, only to find we’ve forgotten the reason when we get there. This isn’t all the time quite the senior moment we laugh about.

From the “Thinking Fast and Slow” notions of Kahneman (2011), blurry thinking is yet another type of thought, often even faster and more penetrating than intuition, yet at the same time working slowly. As you may guess, blurry thinking takes some practice. In a way, it presupposes that you have the tools at hand and only need insight into how to put them together.

3. Preparation. Engaging the Problem

In the preparation stage, the thinker actively engages with the problem. This involves examining known elements, testing initial approaches, and exploring possible structures. The work is often laborious and uncertain.

Here, processes such as reflection, repetition, and trial and error are most visible. Attempts are made, evaluated, and revised. Although progress may appear slow, this phase generates the material required for later insight. It is the disciplined engagement with the problem that sets the stage for transformation.

4. Incubation. The Hidden Work of Thought

Following sustained effort, the problem may be set aside. During this incubation phase, conscious attention shifts elsewhere, but cognitive processing continues beneath awareness. Hadamard emphasized that this unconscious activity is not idle; it reorganizes ideas, forms new associations, and filters possibilities.

The importance of incubation lies in its ability to restructure the problem without the constraints of deliberate control. What appears as inactivity is, in fact, a continuation of the problem-solving process in a different mode.

5. Illumination. The Emergence of Insight

The transition from blur to clarity occurs in the illumination stage. A solution or key idea appears suddenly, often with a sense of immediacy and coherence. What was previously fragmented becomes organized. In more modern terms, this is the time when your thinking becomes self-organized.

This moment is commonly described as insight. However, it should not be interpreted as spontaneous creation. Rather, it is the culmination of prior preparation and incubation. The clarity of illumination reflects the underlying work that has already been done.

6. Verification. From Intuition to Structure

Insight alone does not complete the problem-solving process. The idea must be tested, refined, and expressed in a precise form. This is the role of verification.

During this stage, intuitive understanding is translated into explicit reasoning. Steps are made sequential, assumptions are examined, and conclusions are justified. The solution becomes communicable and subject to evaluation. It is here that focused thought fully emerges.

A significant point is that these four stages of preparation, incubation, illumination, and verification are highly non-linear. At any stage, there may be no pathway to the next stage, and thinking must step back, but with feedback. Multiple repetitions may be required to get anywhere productive. Sometimes, the entire exercise is in vain, with nothing emerging. Such is research and thought.

It is important to listen to Hadamard's ideas. He was an important and highly successful problem-solver, and he is explaining how he does it. His notions of the psychology of problem solving may seem unusual or foreign, but he did it more than once. An important component, it seems, is that when exploring solution ideas at the outset, one should not bring them to a precise form too soon.

7. The Role of Articulation in Clarifying Thought

A critical component of this transformation is articulation. The early stages of thinking are often nonverbal, but clarity requires expression in language or symbolic form. They move from blurry to patterns, and then to writing, diagramming, and structured reasoning, all of which serve to stabilize and refine ideas. The key is to wait until the multiple patterns coalesce into the most likely prospect.

This process aligns with metacognitive awareness, the ability to monitor and regulate one’s own thinking. By articulating thoughts, the thinker converts vague impressions into defined structures, reducing ambiguity and increasing precision. Now you have something to which you can apply solid rigor.

8. Examples.

The quintessential example of this sort of blurry thinking, this time co-present with religious tones, is about Srinivasa Ramanujan, an Indian mathematician (1887-1920). With essentially no training, he produced extremely deep results in higher mathematics. Some are so deep that, to date, no correct proofs are available. Rather than discuss the math, we are concerned with how he did it. He was forthright in telling us. Here are a few highlights.

His own descriptions of his process align closely with Hadamard’s model of invention: prolonged conscious preparation (intense, obsessive work on problems), followed by unconscious incubation (often in dreams or sleep), culminating in sudden illumination where formulas appeared fully formed or nearly so. The “blurry thinking” element, vague, non-verbal, diffuse mental representations that gradually sharpen under sustained attention, maps onto Ramanujan’s intuitive, perceptual style in several ways. Ramanujan repeatedly described mathematical insights arriving as vivid visual phenomena. He produced thousands of results with minimal formal training, often writing down theorems that “were obvious to him” without immediate proofs. He generated families of identities from minimal seeds, relying on an internal sense of coherence and resonance rather than linear logic. Finally, he offered famous dream accounts (formulas revealed by the goddess Namagiri) as textbook examples of Hadamard’s incubation.

In short, while Ramanujan’s visions were often described as strikingly clear upon arrival, the underlying psychology mirrors Hadamard’s “blurry images moving toward focus” exactly.

In another example, one more familiar to most of us, Leonardo da Vinci’s (1452-1519) inventive and artistic thinking aligned closely with the “blurry images moving toward focus” model of creativity described by Hadamard, and even more explicitly than for Ramanujan. Da Vinci’s process was profoundly visual, iterative, and reliant on vague, ambiguous, or diffuse mental and perceptual stimuli that he deliberately cultivated, then refined through sustained attention, analogy, incubation, and eventual crystallization into precise forms (sketches, inventions, or paintings). (Isaacson, 2017.) He developed images, sometimes for years, before completing the final work. After all, imagery was his principal game.  In fact, in his book, Hadamard mentions da Vinci but not Ramanujan. From da Vinci, we have the quote,

“It should not be hard for you to look at the stains on walls, or the ashes of a fire, or the clouds or mud, and if you consider them well, you will find marvelous new ideas, because the mind is stimulated to new inventions by obscure things.”

Ramanujan and da Vinci were both extremely creative geniuses, and by understanding how they think, we gain insight into another facet of genius. Other deep thinkers who recounted these ideas of blurry imagery include Albert Einstein, Hermann von Helmholtz, George Pólya, and Norbert Wiener. In a letter to Hadamard, Einstein wrote that words played no role in his thought processes, which instead relied on signs and more-or-less clear images (Hadamard, 1945). More modern thinkers/creators who used blurry thinking include Paul McCartney (music), Jonas Salk (polio vaccine), and Choreographer Twyla Tharp, just to illustrate the broad spectrum of professions.

9. The Foundational Mechanisms of Transformation

Underlying this progression are the foundational mechanisms discussed earlier: reflection, repetition, trial and error, and the development of reflex. These processes operate across all stages.

• Repetition provides exposure and familiarity.
• Trial and error introduces variation and reveals limits.
• Reflection evaluates and refines understanding.
• Reflex emerges as patterns become automatic through experience.

Together, they form a reinforcing cycle. Over time, deliberate reasoning becomes more efficient, and previously complex tasks can be performed with increasing speed and accuracy.

10. The Value of Blurry Thinking

An important implication of this model is that blurry thinking should not be avoided. In many educational contexts, there is pressure to achieve immediate clarity and correctness. However, this can suppress the exploratory phase essential to deep understanding.

Blurry thinking allows for flexibility, creativity, and the formation of novel connections. It provides the space in which ideas can evolve. Premature closure, by contrast, may lead to superficial solutions and rigid thinking.

 

11. Conclusion

Problem solving is best understood as a progression from vague intuition to structured reasoning. The clarity observed in finished solutions is not the starting point of thought, but its outcome. Through preparation, incubation, illumination, and verification, the mind transforms diffuse impressions into coherent understanding. Even still, and even if you agree to everything above, it is not an automatic transformation to this mode of thinking. It must be practiced, and it can be more difficult for linguistically inclined. On top of that, you must have the time and an environment to do so.  

This transformation is supported by fundamental learning processes, reflection, repetition, trial and error, and reflex, which operate across all stages. Together, they enable the gradual sharpening of thought.

To the teacher whose student said, “It just came to me. I don’t know how,” it may be wise to listen. Young students have not yet transitioned into their fully lingual articulation. Youngsters are still conversant with the brain’s language, seemingly perhaps blurry to us.

In this sense, problem-solving is not merely the application of knowledge but the construction of clarity itself. What begins as a blur becomes focused, not through sudden brilliance alone, but through sustained engagement, disciplined thinking, and the patient refinement of ideas. Ultimately, the movement from blurry thinking to focused thought is not incidental to problem-solving; rather, it is its defining characteristic.

When next you’re working on an important problem, and nothing but blurry images come to mind, it’s just possible your brain is thinking, if only you’d listen.

References

Hadamard, J. (1945). The mathematician’s mind: The psychology of invention in the mathematical field. Princeton University Press*.

Isaacson, W. (2017). Leonardo da Vinci. Simon & Schuster

Kahneman, D. (2011). Thinking, Fast and Slow. Farrar, Straus and Giroux.

Poincaré, H. (1913). The foundations of science. Science Press.

Wallas, G. (1926). The art of thought. Harcourt, Brace and Company*.

 

* Available on books.google.com and other sources.

 

 

©2026

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[1] Both Jacques Hadamard and Henri Poincaré were world-famous (French) mathematicians in their day and are still considered heavyweights years later.

[2] I recall watching a video given by Andrew Wiles, where he described solving the Taniyama-Shimura-Weil conjecture by assembling chunks of other theories, which led to the proof of  Fermat’s Last Theorem.