Seven apples can exist without the word “seven.” The harder question is whether seven can remain seven after the apples disappear.
Imagine seven apples on a table.
You can see them.
You can touch them.
You can move them.
You can eat one.
Now there are six.
For speakers of languages with developed numeral systems, this sounds almost absurdly simple.
But notice what happened inside that simple description.
We treated seven as something that survives independently of the apples.
Seven apples.
Seven stones.
Seven people.
Seven sounds.
Seven days.
The objects change.
The number does not.
Now remove the word seven.
Remove six.
Remove the entire learned counting sequence that allows us to move:
one, two, three, four, five, six, seven…
The apples remain.
Their quantity remains.
But what happens to our ability to represent their exact cardinality?
This is where Pirahã, a language spoken in the Brazilian Amazon, takes us into one of the deepest questions in the study of language and thought.
Not:
Can people see quantities without number words?
Of course they can.
The real question is much more difficult:
What does language allow us to do with exact number once the objects themselves are no longer available?
1. The famous story is “one, two, many”
Pirahã often appears in popular accounts through an unforgettable formula:
one
two
many
It is a wonderful story.
It is also misleading.
Earlier descriptions associated Pirahã quantity expressions roughly with “one,” “two,” and “many.”
But later experimental work showed that these expressions do not behave like exact numerals.
The forms often cited as hói, hoí, and baagiso are better understood as expressions of relative or approximate quantity rather than members of an exact counting sequence.
In experiments reported by Michael Frank, Daniel Everett, Evelina Fedorenko and Edward Gibson, the researchers concluded that Pirahã has no linguistic method for expressing exact quantity — not even exact one.
That changes the problem completely.
We are no longer looking at an unusually short counting system.
We are looking at the possibility of a language without exact number words.
2. “One” and “a small amount” are not the same concept
Suppose I place one fish in front of you.
You use expression A.
Then I place two fish.
You tend to use expression B.
Then I place ten.
You use expression C.
It would be tempting to construct the dictionary:
A = one
B = two
C = many
But now reverse the experiment.
Begin with ten objects and gradually remove them.
If the expressions shift according to relative amount rather than fixed cardinality, the dictionary begins to collapse.
A true numeral such as two should remain two regardless of whether we arrived there from one object or from ten.
Two is not:
relatively few compared with what I just saw.
Two is exactly two.
That distinction is fundamental.
3. Quantity is older than counting
Humans do not need mathematics lessons to notice quantity.
Neither do many non-human animals.
A larger pile differs from a smaller pile.
Two visible collections can be compared.
Small quantities can often be tracked precisely.
Larger quantities can be estimated approximately.
These capacities do not begin when a child learns the words:
one, two, three.
So it would be wrong to say:
Without number words, there is no perception of quantity.
Pirahã speakers obviously inhabit a world containing collections of objects.
They can perceive more and less.
They can interact with quantities.
The fascinating problem begins when we distinguish quantity from exact symbolic number.
4. Seven is not the same thing as “about this many”
Imagine two piles:
● ● ● ● ● ● ●
and
● ● ● ● ● ● ● ●
Without counting, you may perceive that one is larger.
If the difference becomes much greater, comparison becomes easier.
This is approximate quantity.
But mathematics asks another question:
Exactly how many?
Seven is not:
a medium-sized handful.
Seven is not:
a little less than eight.
Seven is an exact cardinal value.
And exact cardinality has an extraordinary property.
It survives changes in the objects.
Seven apples.
Seven knocks.
Seven kilometres.
Seven mistakes.
Seven imaginary dragons.
The category seven is indifferent to what is being counted.
5. This is why the Pirahã experiments matter
The crucial experiments did not simply ask Pirahã participants:
What is your word for seven?
That would tell us about vocabulary.
Researchers instead tested what people could do with sets of objects.
In one kind of task, objects remained visible and could be matched directly.
For example, imagine a row of batteries:
● ● ● ● ● ● ● ●
The participant can place another object opposite each one:
○ ○ ○ ○ ○ ○ ○ ○
No verbal counting is necessary.
You can solve the problem through one-to-one correspondence.
One object here.
One object there.
Pair after pair.
Frank and colleagues found that Pirahã participants could perform exact matching successfully even with relatively large sets when such direct correspondence was available.
That result is enormously important.
It means:
no exact number words ≠ inability to recognize exact correspondence.
6. Now make the objects disappear
Change the task.
Show eight objects.
Then hide them.
Ask the participant to reproduce the same number.
Something important has changed.
The original objects are no longer available for direct pairing.
The participant must somehow preserve:
exactly how many there were.
This is where performance deteriorated.
When tasks required exact quantity to survive memory, spatial displacement or another transformation, Pirahã participants tended to make errors.
They could still estimate.
But exact cardinality became difficult to preserve.
This distinction led Frank and colleagues to a powerful formulation:
number words can function as a cognitive technology.
7. A word can be a tool
We usually imagine technology as something external.
A hammer.
A compass.
A calculator.
A computer.
But a culturally transmitted symbolic system can also function as technology.
Consider what the word seven does.
I show you:
● ● ● ● ● ● ●
You encode:
seven.
I remove the objects.
The visual configuration disappears.
But the label remains available.
Later I ask how many there were.
You do not need to reconstruct the original picture perfectly.
You retained a portable symbolic representation:
7
or:
seven.
The symbol carries exact cardinality across time.
That is an astonishing cognitive achievement precisely because it feels so ordinary to us.
8. Seven can travel
Seven can move from vision to language.
From language to writing.
From writing to memory.
From memory to calculation.
From apples to kilometres.
From kilometres to years.
You can hear seven knocks and later place seven coins on a table.
The sensory modality has changed completely.
The cardinality survives.
This is one reason Frank and colleagues describe exact number language as a cognitive technology for keeping track of cardinality across time, space and changes in modality.
The number word is not merely a name pasted onto an already complete mathematical object.
It can become part of the machinery through which exact quantity is maintained and manipulated.
9. But does that mean Pirahã speakers cannot think about seven?
Now we reach the dangerous sentence.
If we answer too quickly, we will destroy the entire article.
The strongest possible claim would be:
Without the word “seven,” Pirahã speakers cannot have the concept of seven.
The evidence does not justify such a simple conclusion.
In direct matching tasks, participants demonstrated exact one-to-one correspondence for sets larger than the tiny quantities humans can immediately recognize without counting.
So something exact is possible without an exact numeral.
That matters.
The experiment itself undermines the strongest version of:
no word → no concept.
10. But the opposite slogan is also too simple
We could retreat to the comfortable claim:
Words do not matter. People can think anything regardless of language.
Pirahã makes that position difficult too.
Because when exact correspondence could no longer be maintained perceptually, performance changed.
The absence of an exact number vocabulary was not irrelevant.
It appeared to matter especially when exact cardinality had to be stored and transported beyond the immediately visible configuration.
So we need a more sophisticated model.
Not:
language creates all thought.
Not:
language merely labels thoughts that are already completely formed.
But:
language can provide cognitive tools that make some representations easier to stabilize, preserve, communicate and manipulate.
That is a much more interesting relationship.
11. Yélî Dnye gave us one warning. Pirahã gives us another.
In our investigation of Yélî Dnye, we encountered an essential principle:
lexical gap ≠ conceptual gap.
If a language lacks a single ordinary word corresponding neatly to English hand, the physical hand does not disappear.
Speakers can perceive it.
They can interact with it.
Relevant conceptual distinctions may be available through other structures and contexts.
Pirahã now forces us to add a second principle:
lexical gap ≠ conceptual gap — but lexical tools can still change what is easy to do with a concept.
These two statements do not contradict each other.
Together they give us a much stronger theory.
12. An apple is not a number
Take seven apples.
Now remove the apples and keep seven.
What remains?
Not colour.
Not shape.
Not weight.
Not taste.
Not position.
Seven is an abstraction over collections.
This becomes clearer if we compare:
seven apples
seven people
seven seconds
seven ideas
There is no sensory property shared by an apple, a second and an idea that looks like “seven.”
Exact number therefore gives us a way to preserve a structural relation while discarding almost everything about the objects themselves.
That is why symbolic number is so powerful.
13. The child who learns “one, two, three” is not merely learning vocabulary
Children often learn to recite a counting sequence before they fully understand what its words mean.
They can say:
one, two, three, four, five…
almost like a rhyme.
But eventually something extraordinary happens.
The sequence becomes connected to exact cardinality.
The child discovers that the last number reached while counting a set tells how many objects are in the set.
Then another insight develops:
if you add one object to a set of seven, the next number in the sequence gives eight.
The verbal sequence has become more than memorized sound.
It has become an algorithm.
14. This is Thinking Instead of Memorizing — with a twist
Normally we oppose understanding to memorization.
And rightly so.
But the history of numerical cognition shows something subtler.
Sometimes a memorized symbolic sequence can become raw material for a much deeper conceptual system.
A child may initially memorize:
one, two, three, four…
without understanding cardinality.
Later, the structure of that sequence participates in mathematical thought.
So memorization is not always the enemy.
The problem is memorization without reconstruction of the mechanism.
A list can become a tool.
But only when the learner understands what the tool does.
15. “Two apples minus one apple” contains hidden machinery
A familiar arithmetic exercise says:
You had two apples. One was eaten. How many are left?
One.
For a numerate adult, the answer is instantaneous.
But look at everything hidden inside the task.
The learner must understand that:
two describes cardinality rather than appearance;
removing one object transforms the cardinality predictably;
the result can be represented independently of the remaining apple;
and the same operation applies to:
two apples,
two stones,
two cars,
two planets,
or two imaginary objects.
The apples are almost irrelevant.
They are props for an abstract operation.
That is why somebody can understand apples perfectly well and still fail to understand the mathematics represented through them.
16. Arithmetic is not hiding inside the fruit
This is where the apparently comic classroom situation becomes philosophically useful.
A teacher points to physical objects because they seem concrete.
But concreteness does not automatically generate abstraction.
You can eat an apple without understanding subtraction.
You can cut an apple without understanding fractions.
You can arrange seven carrots without possessing a symbolic system for seven.
The physical world supplies instances.
Mathematics supplies relations among abstractions.
Language can help build the bridge between them.
17. Exact equality is stranger than it looks
Consider:
● ● ● ● ● ● ●
and
▲ ▲ ▲ ▲ ▲ ▲ ▲
Visually, these sets are completely different.
Different objects.
Different shapes.
Different arrangement.
Perhaps different sizes and colours.
Yet mathematics declares:
7 = 7
It ignores almost everything perception gives us.
That is an extraordinary cognitive compression.
We preserve only cardinality.
A number word gives that preserved property a reusable symbolic handle.
18. Approximate number and exact number are different achievements
Humans possess systems for approximate quantity.
We can often judge that a crowd of eighty people is larger than a crowd of forty without counting everyone.
Approximation becomes harder as quantities get closer.
Exact symbolic number behaves differently.
81 and 82 are perfectly distinct even though visually estimating two crowds of those sizes would be extremely difficult.
The difference of one remains exact.
This precision does not come from sharper eyesight.
It comes from the symbolic numerical system.
19. A calculator is not proof that you cannot think
Suppose I ask:
7,846 × 293
You use a calculator.
Does the calculator create multiplication?
No.
Does using it mean you lack mathematical concepts?
No.
But does the tool dramatically expand what you can reliably compute?
Obviously.
This gives us a useful analogy for number words.
Calling them a cognitive technology does not mean the unaided human mind contains no quantity concepts.
It means symbolic tools can extend cognitive performance.
A compass does not create space.
Writing does not create memory.
A calculator does not create quantity.
And number words need not create perception of numerosity in order to transform what humans can do with exact numbers.
20. Language can externalize precision
Suppose I tell you:
Remember 47.
Two seconds later, I ask:
What number did I say?
The symbolic label can be rehearsed.
Forty-seven.
Forty-seven.
Forty-seven.
Now imagine retaining an exact visual collection of forty-seven unrelated objects without counting or naming the quantity.
That is a very different cognitive problem.
A numeral compresses a large set into a compact symbolic form.
47
is extraordinarily cheaper to maintain than a detailed visual representation of forty-seven individual objects.
Language has compressed cardinality.
21. Writing compresses it again
Spoken number:
seven thousand four hundred and eighty-six
Written numeral:
7,486
The representation becomes even more compact.
Then mathematical notation allows:
7,486 + 319
The original objects have vanished completely.
We can manipulate cardinalities without seeing anything being counted.
Human numerical culture builds layers of cognitive technology:
perception → quantity → counting words → exact cardinality → written numerals → arithmetic notation → mathematics
Each layer creates new possibilities without requiring us to claim that the previous layer contained nothing.
22. Pirahã is not a “primitive language”
This must be stated explicitly.
A language lacking exact numerals is not an incomplete language waiting to become English.
Languages develop resources according to histories, practices and communicative environments.
Pirahã is a human language capable of human communication.
Its numerical vocabulary tells us something fascinating about one particular domain of symbolic representation.
It tells us nothing about the intelligence or human worth of its speakers.
Nor should Pirahã be treated as a fossil from some earlier stage of human evolution.
Its speakers are contemporary human beings.
The linguistic system is interesting precisely because it shows how much variation human cultural systems can sustain.
23. Nor does Pirahã represent “the Amazon”
Amazonia contains enormous linguistic diversity.
Number systems vary widely across its languages.
Some Amazonian languages possess richer numeral inventories and counting practices.
So:
Pirahã → all Amazonian languages
would be as absurd as:
English → all European languages.
A remarkable linguistic system should make us more precise, not encourage another stereotype.
24. What happens when a numerate culture arrives?
The story becomes even more interesting when speakers encounter schooling, money, trade and surrounding languages with developed numerical systems.
Numeracy is teachable.
People can learn Portuguese numerals.
They can learn arithmetic.
Nothing about Pirahã biology prevents this.
This again demonstrates why we must distinguish:
the resources conventionally available in a language and culture
from:
the ultimate cognitive capabilities of individual humans.
Languages provide tools.
Humans can acquire new tools.
25. So can you think about seven without the word “seven”?
We can now answer the title more carefully.
Can you perceive a set containing seven objects?
Yes.
Can you establish one-to-one correspondence between that set and another visible set without saying “seven”?
Yes.
Can you recognize approximate differences in quantity?
Yes.
Can exact cardinality become much harder to preserve when the objects disappear and no exact numerical symbol is available?
The Pirahã evidence strongly suggests that it can.
So the interesting boundary is not:
thought / no thought.
It is between different operations:
perceiving
matching
estimating
remembering
symbolizing
communicating
calculating
They are not the same cognitive act.
26. This changes the language-and-thought question
The traditional debate is often framed badly:
Does language determine thought?
Yes or no?
Pirahã suggests a better set of questions.
What representations are available without language?
Which representations can be maintained without symbols?
What does naming make portable?
What does a counting sequence make computable?
What can be perceived but not reliably stored?
What can be stored but not easily communicated?
What becomes possible when a cultural symbol system is learned?
Now “language and thought” stops being a philosophical slogan.
It becomes a series of testable mechanisms.
27. Words do not merely describe finished thoughts
This may be the most important lesson.
Sometimes a word names something we already perceive easily.
tree
red
hot
Sometimes a linguistic distinction organizes relationships.
Sometimes grammar requires us to encode information.
And sometimes a symbolic system gives us a stable object with which cognition itself can operate.
Seven is not simply a label attached to seven visible things.
Once acquired as part of an exact counting system, it becomes something we can carry into situations where those things are absent.
We can add it.
Subtract it.
Multiply it.
Remember it.
Write it.
Transmit it.
Build mathematics from it.
The word has become a tool.
28. Our map has just become more complicated
Maniq showed us that languages can build very different lexical resources for sensory categorization.
Yagua showed us that temporal distance can become part of grammatical tense.
Guugu Yimithirr showed us that spatial description can begin from different coordinate systems.
Pirahã now adds something different again.
Language can be more than a way of classifying experience.
It can become part of the technology through which an abstraction is preserved and manipulated.
That is a much stronger claim than “different languages have different words.”
And a much weaker claim than “language imprisons thought.”
It occupies the interesting territory between them.
29. Perhaps the real invention was not the number
Seven objects existed before anybody counted them.
Seven stars could appear in the sky before anyone named seven.
Seven footsteps could be taken before any language possessed a numeral corresponding to seven.
So humans did not invent the quantity.
What cultures invented were systems for making exact quantity portable.
Counting sequences.
Number words.
Tallies.
Numerals.
Arithmetic notation.
These systems allow an invisible abstraction to survive the disappearance of the objects that instantiated it.
That is the technological achievement.
Conclusion: seven is easy while the seven things are still there
Place seven stones on a table.
You may not need a number word to match them with seven shells.
The stones themselves can carry the information.
Remove the stones.
Walk away.
Wait.
Now reproduce exactly the same quantity.
The problem has changed.
Something must carry seven after the seven things are gone.
A developed numerical language gives us one extraordinarily powerful carrier.
This is why Pirahã matters far beyond an exotic story about a community supposedly saying “one, two, many.”
The deeper lesson is not that people without exact numerals cannot perceive quantity.
Nor is it that language creates mathematics from nothing.
It is that human beings build symbolic technologies that allow cognition to do things perception alone does poorly.
A number word can preserve exactness.
A counting sequence can organize it.
Writing can externalize it.
Mathematics can manipulate it after every original object has disappeared.
Seven apples can exist without seven.
But once the apples are gone, seven has work to do.
A word does not have to create a concept to become one of the tools with which we learn to carry that concept beyond the world in front of us.
Tymur Levitin
Founder & Director, Levitin Language School
Certified Translator and Language Teacher
© Tymur Levitin / Levitin Language School. All rights reserved.
