Our Gemara on Amud Aleph describes a certain saddle used for women, known as a Kumni, which presumably functioned in a manner that allowed a woman to ride side-saddle and more modestly. 

This brings to mind the famous Rashi (Shemos 28:4) that describes the form of the Ephod. The Ephod was one of the garments that the high priest wore, composed of cloth and had the breastplate attached to it. Apparently, there is no explicit teaching about what this item of clothing looked like. Rashi describes it as similar to a riding apron worn by women of his time. One can imagine the function of an apron shaped garment to preserve the modesty of the woman rider, without her having to wear pants. How Rashi comes to this idea is significant.

He introduces his thought and description of this vestment by saying, “Though I have not heard a teaching or tradition that describes this garment, my heart tells me it is shaped as follows…” Rashi is reporting that he had an intuition and insight into the Ephod’s shape, based on having seen a woman’s riding apron.

Rashi’s experience is reminiscent of a number of great thinkers and scientists who described their breakthroughs as coming from imaginative visions. For example, Dimitri Mendalev is said to have first seen the Periodic Table in a dream, and August Kekule saw the Benzene molecule in a daytime reverie, while Einstein described himself as imagining riding a light wave. (Lois Eisenman, Perspectives in Biology and Medicine, Spring 1997, 40, 3 “Toward an Understanding of Intuition”.)

But there is a second time that Rashi uses the phrase, “My heart tells me” in his discussion in Shemos. The second usage is perplexing. He says, “My heart tells me it is an article of clothing based on the following verses…” If Rashi had proof texts that it is a kind of garment, and indeed if you look up his sources they are compelling, why then did he say “my heart tells me?” This second usage seems unrelated to intuition, as he had ample proof. 

To answer this, we must study the phenomenon of intuition. We can define intuition as a process by which the symbolic part of the mind discerns a pattern or meaning in totality, without a linear step-by-step deduction. Our consciousness is divided between things we isolate by intense focus and scrutiny, versus background noise. Consider the example of a party where there is a cacophony of noises and conversations which your ears hear, but your mind only processes deeply what your friend is telling you. Still, all of the sudden, if someone mentions your name across the room, your ears perk up. A less conscious part of your mind must have been sorting out and listening to all the noises, and it only flagged the noise of relevance, when your name was said. Since this part of the mind is crunching much larger amounts of data, it has to rely on a different mechanism than scrutinizing each stimuli in detail. It must rely on some kind of data sampling method, which lacks precision but gains breadth to allow for filling in the blanks with suppositions. 

So too, even with ideas, a part of the mind may have grasped a pattern but your conscious thought might only notice it via a symbolic image, thought or feeling. We all have had the feeling that we are being followed, even though we do not have eyes in the back of our heads. Somehow, a collection of subtle cues (echoes, shadows, reflections, or the way in which local animals slink, vegetation rustles, or even an uncanny quiet) signal to us that something is off, just beneath our conscious perception. 

Similarly, everyone instantly knows that a circle cannot be a square, though it might take time to construct a logical proof that this is so. What is happening cognitively? Apparently, there must be enough general data to strongly support the truth of this, but there is not enough linear step-by-step proof. The mind uses general points to identify the overall pattern. This is also why we can understand God through what’s known as Via Negativa. It's a way to understand God by focusing on what God is not, rather than what God is. We can never understand what God really is, but we can understand what He is not.  We know he is not physical, not weak, not unwise, not uncompassionate. When we do say he is wise or compassionate, those are inaccurate borrowed human terms.  If God were compassionate, he would then be subject to physical bounds, as something (His emotions) would influence Him. Only physical objects change, which makes them subject to time and entropy, and having a beginning and an end. Even to say God is mighty or wise is not accurate, because wise means to “know something” or to be strong “means to contain a certain power”. But if it is something to have or to contain, then it is a quantity, and therefore physical and not infinite.  God’s wisdom and power is infinite so calling Him wise or powerful is not anything like human wisdom or strength. All this is explained in Rambam Yesode HaTorah, Chapter One.  Yet, by knowing what God is not, we get an intuitive understanding of what He is, just as we know about the square and the circle, even though we did not fully connect the dots.  

This is how intuition works. Without fully worked out logic, the quick and general processor in your mind picks up a pattern which then becomes subject for possible rational analysis. Now we can understand Rashi’s choice of words for his second assertion. True, he found a proof text that the Ephod was an article of clothing, just as Einstein ultimately did the math to prove his intuition, but Rashi realized that the inspiration to notice those verses and the ability to link them together to prove the point came from his intuition.