The Cognitive Origin of Numbers

Numbers. We typically think of them as objective symbols that never change. 4 is always smaller than 5. 5 is always smaller than 6 and so on. But that’s not actually the case. If you have two numbers the larger number can sometimes feel subjectively smaller. Numbers are inherently fuzzy. And more importantly, this word “fuzzy” has a dual meaning. In order to learn abstract concepts like numbers we needed to construct these concepts from tangible sensory experiences. So today, whenever you think of a number you’re not just activating some abstract symbol. Your brain is activating a hidden foundation of sensory elements. And in this video, we’ll peel back those layers so that you can see this hidden sensory foundation. And I’ll show you how this architecture in your brain is distorting your perception of numbers on a day-to-day basis. And we’ll use the consistent example of prices. 1. Size Quick. Answer yes or no. Is 1 less than 5? Is 4 less than 5? If numbers were truly discrete symbols in our brain then you should be able to answer those questions with equal speed. Yet people take longer to determine that 4 is less than 5. Even though your conscious brain is seeing these numerical symbols unconsciously your brain is activating a hidden foundation of spatial size. So here, the spatial magnitude of 4 looks very similar to that of 5. And thus, it takes longer for your brain to distinguish these numerical symbols. And same with these questions. Is 2 less than 4? Is 26 less than 28? Here, people take longer to determine that 26 is less than 28 because again your brain is activating spatial size. A change from 26 to 28 isn’t that noticeable but a change from 2 to 4 is very noticeable It’s double in size. And because of that hidden foundation of spatial size you’re more likely to buy products when prices are visually smaller because your brain confuses that visual size with numerical size. Now those examples involve spatial size, but the concept of size is more schematic in nature. For example, how would you convey a large size to somebody? It was huge. Or… It was hu-u-u-u-u–u-u-u-ge. In order to convey a large size to somebody we intuitively enlarge the size of the linguistic symbol. But how does this involve prices? Well, compare these prices. They’re exactly the same except one has a comma. Yet people perceive this price with the comma to be larger. Why? Sure the comma extends the surface area. But more importantly, people conceptualize this linguistic sound as “one thousand two hundred fifty” which is 8 syllables. Without the comma, it becomes “twelve fifty” which is 3 syllables — and a much smaller linguistic size. And again, people’s brains are confusing the size of that linguistic sound with the numerical size of the price because of that hidden foundation of sensory size. 2. Distance We also conceptualize numbers along a horizontal spectrum. And the particular directionality of that spectrum will depend on your language. If you read from left to right then you conceptualize numbers as getting larger toward the right. But if you read from right to left then you conceptualize numbers as getting larger toward the left. I refer to that concept as canonical motion in The Tangled Mind. But for our purposes, because of this inherent foundation of spatial distance a discounted price will seem more enticing when it appears spatially further from an original price With that physical gap, your brain will perceive a larger numerical gap. In fact, specific digits can trigger this effect. In the Mueller-Lyer illusion the bottom
line seems longer even though both lines are identical because of these endpoints that are asymmetrical and pointing outward Because they’re pointing outward, your eye needs to travel a larger distance from endpoint to endpoint. And because this effect is stemming from the asymmetry of these endpoints we can replace these endpoints with something else, like digits. In one study, a discount from 7 to 5 seemed more enticing when the 7 was facing the left because both digits were then facing outward so it replicated the same effect where it created this visual gap which tricked their brain to perceive a larger numerical gap. 3. Motion In a second, you’ll see a box moving across this screen. Now, if you had to guess, where exactly on the screen did that box disappear? Most people guess somewhere around here. But it was actually here. People’s estimates are influenced by the momentum of that box. Here’s a typical response pattern. You can see forward displacements along all directions. And you can even see the influence of gravity. When the box is moving up, people estimate shorter displacements. When the box is moving down, people estimate larger displacements. More importantly, we inject this sensory concept of momentum into abstract concepts. When you’re riding a train and you reach a station that is midway to your destination your destination will feel closer to you because you’re simulating your own momentum through space. And this effect also happens with time. Since time is abstract, we used motion as a building block to construct our understanding of time. If there’s an event in the future and an event in the past that are both equidistant from you you perceive the future event to feel subjectively closer because again you’re simulating your momentum through time so you exhibit that same forward displacement And it’s a really interesting concept and there are so many other fascinating examples And I’d really encourage you to read The Tangled Mind to learn all of those other examples. But for our purposes, we inject this same sensory concept of momentum into the numerical domain. In one study, there was a ranking of colleges. And a college seemed better if it moved from 6th to 4th compared to a college that moved from 2nd to 4th. So even though both groups saw 4th place that’s not how their brain was conceptualizing and evaluating that number. With the ascending college, again people simulated the momentum so they conceptualized a number and ranking that felt subjectively better than 4th place. And you can see the same effect with addition and subtraction. Research shows that we overestimate addition problems while we underestimate subtraction problems. Why? Well, if you’re adding two numbers, then an entity is getting larger. And because of this growing size, your brain exhibits a forward displacement so that it conceptualizes an entity that is even bigger. And the reverse happens with subtraction. If something is shrinking in size, then here, the forward displacement would be even smaller in nature. And we naturally extend this concept with prices. And I would argue that this effect has been a hidden mechanism inside the anchoring effect. At the beginning of his presentations, Steve Jobs would often display a very high price. But then later, he would change
that number to a much lower price. So even though the final number was still a very high price the new price just seemed much lower because of the relative comparison. The anchoring effect is very well known but I think marketers and psychology researchers are missing a hidden effect inside here. So the relative comparison does absolutely play a role but we also need to consider the inherent effect of momentum. By starting with a very high price that transforms into a much smaller price your brain is conceptualizing this change as a motion of reduction. Therefore, the new price — whatever it is — it will feel subjectively smaller because you’re simulating the inherent momentum of that change. And we need to distinguish this separate mechanism inside the anchoring effect because there are certain variables that could change the inherent effectiveness of this concept. For example, when you see infomercials you typically see multiple price changes. Get this great product for $99. Then $79. Then $49. Instead of a single change from $99 to $49 perhaps multiple price changes in between reinforce in our brain a stronger concept of motion. And because of that stronger concept of motion it strengthens this inherent effect of the simulated momentum. And this effect is likely happening in other contexts. Compare these prices: $29 for 70 items vs. 70 items for $29. People perceived the second framing to be a smaller price. When a large quantity appears first you’re moving from a large quantity to a smaller quantity. So here, there’s downward momentum and motion. So your brain conceptualizes a number that is even smaller in nature. Or consider price ranges. A business might say “Well, it’ll cost between $10k to $20k” Here, momentum is getting bigger. So your brain conceptualizes a price range that feels subjectively larger. If marketers subtly flip that directionality, “Well, it’ll cost between $20k – $10k” This framing moves from large to small. So here, your brain might conceptualize a final price range that feels subjectively smaller. 4. Weight Look at these letters and numbers. Everything looks pretty even and symmetrical, right? Well, when you flip them upside down, you’ll realize that the bottom halves were actually thicker. Designers add more visual weight toward the bottom of stimuli in order to compensate for how our brains actually perceive visual stimuli. Thanks to gravity, we expect things to fall downward. So we intuitively expect more visual weight in those areas. Sure enough, in one study, cookies seemed lighter toward the top of a package as if they were diet cookies. Those same cookies seemed heavier toward the bottom. Therefore, numbers toward the bottom of stimuli will likely feel heavier and thus larger. Numbers toward the top will probably feel lighter and smaller. 5. Location Similar effects happen along the horizontal spectrum. Cookies on the left seem lighter, while cookies on the right seem heavier. Here, this effect is happening because we conceptualize this rectangular frame as a fulcrum. Quote: “Because our eyes enter a visual field from the left… …the left naturally becomes the anchor point or visual fulcrum… …the further an object is placed away from the left side, the heavier the perceived weight.” Now, based on that reason, you might think that numbers on the left will seem lighter and thus smaller. But here, I think that we’re seeing a competing mechanism. First, you should know that our brains are constantly simulating interactions in the environment. If you’re right-handed, then you prefer a mug where the handle is on the right because you can more vividly simulate yourself reaching and grasping that handle. Now, depending on the physical size of an object, we use different grips. If we’re holding an apple, we use what’s called a power grip. If we’re holding a cherry, we use what’s called a precision grip. Those concepts involve physical size but remember, our concept of numbers are comprised of physical size so we still perform those motor simulations with the abstract concept of numbers When people see small numbers, they’re quicker to perform a precision grip. When they see large numbers, they’re quicker to perform a power grip. So even though numbers are intangible they’re built upon a foundation of physical size and so it still triggers that same motor simulation. Now, how does this involve location? Well, imagine a number on the left and imagine a number on the right. Here, not only is your brain activating physical size, but it’s also activating a horizontal spectrum like we discussed earlier. With this number on the left, you have this boundary that is blocking any leftward motion. So if you are mentally interacting with this number which, again, you are then the only direction in which you can move this number is toward the right which, based on your conceptualization, will become a larger number. When a boundary appears on the right, however, this number can’t get larger. Here, your brain can only move this number in a leftward direction which is the smaller side of the numerical spectrum. So from your brain’s perspective, a price on the right side will feel more intuitively appealing because that number can only get smaller. 6. Object Here, we have nine separate dots, but your brain sees three entities. Your brain groups these dots as
a unit so it neglects the visual boundaries of these individual dots. On top of my fridge back there, you can see boxes of food. So if you were to see a box of oatmeal next to a box of cookies if your brain is visually grouping those boxes then your brain is also conceptually grouping them. The cookies will seem healthier because it will inherit healthiness from the oatmeal. The explanation is beyond the scope of this video, but it involves feature integration theory and the process in which we extract and merge features into our perception in a way where there is cross contamination. But for our purposes, the same effect happens with numbers. In my video about hidden cues in Amazon I described a study where people were more likely to buy a skate when the price was adjacent to the words “low friction.” Because of that visual proximity people’s brains merged the concept of low with that price. So the price seemed lower and more appealing. And then I explained how Amazon puts small numbers near their prices so that you visually group these stimuli and you perceive the actual prices to be smaller because of that convergence. 7. Orientation So based on our evolutionary history our brains are hard-wired to follow people’s gazes. So if we see somebody looking in a particular direction then we instinctively look in that same direction. But more importantly, research seems to suggest that we overextended this deeper mechanism. For example, people indicated how much they would pay for a product by sliding a marker to the right. And when that product was facing right people indicated a higher amount because of their heightened attention toward that directionality. When we see a shoe facing a particular direction our brain is partly conceptualizing that shoe as an anthropomorphic entity. And so whatever direction the shoe is facing it activates the deeper hardwiring in our brain that causes us to look in that same direction. And that might sound weird and far-fetched but when you look at our language you start uncovering these humanlike metaphors, like front and back. We can refer to the front of a house or the back of a house. Those words are so pervasive that they don’t feel like metaphors. But we derive these concepts of front and back from our own human bodies. We have a front side that we project to the world. And we have a back side that we can’t see — that kind of remains hidden. If we, as humans, had eyes all around us and if we just had this global sense of sight and if we could see behind corners then we wouldn’t have the concept of front and back to then apply to external objects. Our own anatomy and biological structure is helping us to construct our understanding of the world around us and including the concepts that we come to know. But how does this involve numbers? Well, just like we attribute a front and back to external stimuli like houses we do the same with digits. All of the digits from 0 to 9 are facing a particular direction — either the left the center or the right. And if we are truly conceptualizing these digits as anthropomorphic entities then whatever directions those digits are facing they should activate the same hardwiring in our brain that causes us to push our attention toward those directions. And that’s exactly what happens. Consider these prices: 58 and 98. Let’s forget the 8 because it’s facing the center in both prices. But the 5 is facing right, while the 9 is facing left. When you see 58 because the 5 is facing right you allocate your focus toward the 8 in this
price. And because you’re focused on the 8, you’re more likely to round that price up to 60. But with 98… Here, the 9 is facing left. So you allocate less focus toward the 8 in this price. And because you’re not allocating as much focus on the 8 the price feels subjectively closer to 90 because you’re rounding down. And I assume that this effect would strengthen the effectiveness of charm prices. $2.99 vs. $3.00 is only a 1 cent difference but from your brain’s perspective it feels like a 1 dollar difference because your brain starts encoding the magnitude of this price as soon as your eye hits that first digit. And then you devote fewer and fewer resources to the remaining digits as you scan across this price. So if this initial digit is facing left, then it’ll have the opposite effect of the shoe study where it kind of acts like a break and it’s pushing your attention away from the final digits and so you’re more likely to further neglect those final digits in the price. But here’s a question. What specific traits determine the direction of a digit? The center is pretty easy because both 0 and 8 are symmetrical so it’s like they’re facing the front. But let’s look at this 5-digit. This written numeral is just a made-up symbol. So why does it feel so natural to assume that this side is the front, while this side is the back? Why couldn’t this side be the front? Somebody had asked me that a while ago, but I didn’t have a good answer at the time. But here is what I think is happening. Remember, I argued that we derive the concept of front and back from our own anatomical structure. So when we see a digit, we’re engaging in a process that I sometimes refer to as projective simulation. You’re mentally immersing your body into this digit. So if you were to see the digit 5, how would you need to immerse yourself? Well, your head would be here. Your back here. Your legs here. Try to immerse yourself in the opposite direction. You can’t. Your legs don’t physically bend that way. With a left facing digit — like 2 — a human representation would be facing the left. And with the center digits, this symmetry matches the symmetry of a human body. So whenever you see a digit, you’re unconsciously simulating this digit as an anthropomorphic entity. So if that digit is facing a particular direction you intuitively allocate more attention toward whatever direction that digit is facing. Alright, so those were some of the sensory elements that are hidden inside our concept of numbers. The main takeaway of this video is that numbers are less definitive than they seem. When you peel back the layers of how our brains actually conceptualize numbers you find that certain changes to the visual display of a number can trick our brain into perceiving that number differently. And these changes influence our perception because we constructed our concept of numbers from bodily and sensory experience. Today, when you think of a number you’re not activating an abstract symbol you’re activating sensory concepts. And that, my friend, is why all numbers are inherently fuzzy. Pun intended. Alright, so where to go from here if you want to learn more? This was a long video, but there is so much more information that I couldn’t fit. If you want to learn more pricing applications you can go to my website and I have a very long detailed guide which explores many other interesting principles of pricing psychology. It’s a free download. And you can download many other guides as well. Or if you want to learn more about this core idea of hierarchical learning about how we use sensory concepts as frameworks to construct abstract concepts which is a really interesting idea with many other applications outside of numbers I’d really encourage you to read The Tangled Mind which goes into so much more detail on how this actually occurs and the practical implications of this mechanism. It’s on Amazon, Audible, and all of the major places. And I’ll see you in the next video.

12 thoughts on “The Cognitive Origin of Numbers”

  1. In this video, I argue that sensory concepts constructed our understanding of numbers. Here are the timestamps for each sensory concept:

    0:55 Size
    3:06 Distance
    4:33 Motion
    9:41 Weight
    10:24 Location
    12:51 Object
    14:09 Orientation

  2. Thanks for your fantastic insights, as always, Nick! Wanted to share this A/B test case study, as it helps support some of what you've said:

  3. fantastic nick !!! minor aspect in terms of implemention. but huge impact. i have applied a few of these while presenting budgets to management as plan 1 or plan 2, with figures to influence one of these two plans. also the design of table, column width and alignment will indirectly influence to choose the one that i prefer !!!! the orientation of figures helped me a lot.

    great nick ! thank you !!

  4. I love these videos but I can't help but feel theres a long list of pros and cons for every application of pricing and a piece on incredible content would be a chearsheet on how to get the pricing right for a product based in a these studies.

  5. This Video is INSINELY AWESOME! Great Info and great explanation nick! Loved it! By the way, do you have any of your books in spanish? Great video Nick, great video…

  6. This is completely unrelated to this video but I finding myself wanting to clear my throat when I watch ur videos. I think it has to do with your voice.

  7. Why does this video not have a million likes? I was so dumbfounded by this information that I immediately went and got the Audible version of your book. Can't wait to start it.

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